Are you using the Kingdom Building rules from Ultimate Campaign or from the AP? Reason I ask is that the AP rules are really less than ideal because, as you say, they actually do encourage a rather small kingdom.
Anyhoo, as to your questions, there are a few ways to go about it.
First, you can go ahead and start book 3 anyway. Sure, they may not have claimed every hex between Oleg's and Varn, but they can still go and explore them. It's always subject to GM's discretion anyway.
Second, you could run a fey-themed module set in the PC's kingdom to give them some more taste of the overall plot.
Third, do away with the hexploration. By this point in their careers the party has a) spent a heck of a lot of time on hexploration and b) are sufficiently important enough in their kingdom that they would be unlikely to do the actual exploring themselves rather than hire it out. One of the two main problems with the AP is the massive amounts of hexploration which will no doubt get tiring (the other being the rather thin plot threads holding it together if the GM doesn't add details to help tie it all together.
Fourth, consider running the kingdom in the background. You didn't really state why they haven't done more, but you have eluded to part of the problem when you say several of the players will be bored. By running the kingdom in the background, you can have the kingdom grow without having to deal with all the details at the table.
Now some questions of my own:
1. How much foreshadowing have you put into the AP? Have the PCs met Varn yet? If not, the disappearance isn't likely to really affect them much. Likewise, have you given much in the way of hints about N? If the answers are No to both of these, then I really suggest adding in some content (either another fey-themed module) or just some plots of your own to help introduce these elements.
2. Have you considered mixing some of the elements of the books up a bit? Especially if your party hasn't really interacted with Varn yet, it might be a good idea to hold the Rushlight Tournament now. Have Irovetti send them an invite and they can go there and meet all the notable figures in the region, which will also help set up the rest of the books. Note, you do not have to have Irovetti actually attack the PCs kingdom yet. This could be seen as a means for him to judge the new local lordlings in his neighborhood. Even better, allow the PCs to snoop a bit and maybe they unearth some research that Irovetti has been doing into liches or cyclopses, etc. Maybe even find a love letter between him and N, etc. In other words, you start laying the groundwork for both I and N to be the major players they are, etc.
3. Finally, why hasn't the party expanded more? I addressed this a bit above, but again, if the party just isn't into the kingdom building phase, then just run it in the background and let them get on with the fun stuff.
Just a few thoughts/ideas for you. I'm sure others will be along shortly to give even better ones. :)
Ok, sounds like you actually have a decent handle on things. You've already got them on a better footing than the AP would put them based on foreshadowing, etc.
One option you could consider would be to have the kingdom turns take seasons as opposed to months. You could, in that way, do a few kingdom turns and cover a lot of "calendar time" without taking up a lot of table time. Obviously though you know your players and it sounds like you are doing a good job of making sure things don't get too boring for anyone. If you do go the "season" route instead of the monthly, you could then allow more to happen per turn to account for the longer time.
As to how to encourage the party to expand, I agree that its a delicate issue. Its true that if their consumption is good, then there's no real need for them to expand other than ego. As pennywit says though, just because the PCs are not expanding their kingdom, doesn't mean that others are not either. Drelev in particular seems like a good candidate to try to encroach. Varn may or may not based on how good the relations with the PCs are. An additional idea might then also be to let word slip to the PCs that Varn and Drelev have been invited to the Rushlight Tournament when the PCs have not. If they try to enquire as to why, have it revealed that all the "notable" lords, nobles, and kingdoms in the area were invited. As you say, the PCs kingdom is too small to attract Irrovetti's attention at this point. More to the point though, there's nothing a player hates more than to think that his or her PC is merely ordinary and not noteworthy.
In my game, Varnhold starting claiming hexes west of the mountains (as their eastward expansion is stymied by the centaurs), which caused my players to expand towards the mountains themselves before Varnhold could take that territory.
I also offered incentives for building a road from Oleg's to Mivon, their neighbor to the south, which caused them to expand in that direction as well.
I kinda did #3 myself, although we played boardgames and some short other games. I meant to run some kingdom turns while we were doing that to spread them out, but that idea somehow fell by the wayside. So when we meet tonight, we're going to advance the timeline a few years before starting book 3. I've come up with some plots and events to unfold over that time so it isn't just processing kingdom turns.
There are some solid suggestions here already. The idea that the land on their map is not specifically reserved for them, and if they don't claim it others might, should be a powerful incentive to expand. It doesn't even have to be other nations claiming their hexes: if the PCs clear a piece of prime real-estate such as the Forgotten Keep or the Stag Lord's fort of its inhabitants and then leave it abandoned, something else will take up residence. You can just roll on the wandering monster table to see what moves in. You could even do it in waves - keep rolling on the table each month to see which new monster(s) has stumbled across the place, and decide how they interact with the current inhabitants (The newcomer is driven off? They team up? One tames or subjugates the other? The newcomer drives off or kills the incumbents?)
Another incentive nations have historically had to expand is a need for Lebensraum. If the players have a lot of new citizens arriving from Rostland or the River Kingdoms, they might need to expand just to accommodate the influx. The rules as written don't really work like that (citizens just seem to appear when the players build structures for them), but you could have custom events or even just do it narratively. They players could also get word of immigrants from the south being attacked and killed (or enslaved) en-route to the new barony, hopefully inspiring them to expand south to secure the immigration route. Immigrants and refugees might also just build small villages in the Greenbelt outside of the players' boundaries, Tatzlford-style, and then come and beg for protection when they have problems with marauding monsters. Also, perhaps nobles and knights and other powerful individuals from elsewhere deign to move into the players' new territory, but expect to be parcelled out land and settlements of their own to hold in the players' name.
Part of the problem is that you need months of game time to go past for their realm to expand, but back-to-back kingdom turns aren't really very interesting. If you drop in whole modules (such as Realm of the Fellnight Queen), the players can spend a lot of playing time on them with only a few days of game time going by, which doesn't really help rack up the kingdom turns.
What you really need are some ongoing slow-burn things happening in the world that give the players something relatively quick but interesting to do each month at the character scale between kingdom turns. They can then do a kingdom turn, play character-scale for 10-30 minutes, do another kingdom turn etc.
The players could consciously choose to approach any remaining quests and monsters on their map that way; instead of sweeping through the whole area in a continuous expedition, they might just deal with one thing, then head back to town and wait a month before doing the next. Turn the five minute working day into a five minute working month, as it were :)
Heating up the tensions in Brevoy might also give you some mileage... not necessarily to the point of open civil war, but enough to send a flood of refugees into the Greenbelt. Responding to reports of the activities of the various houses from the north and perhaps sending off spies and agents of their own could be quick and interesting, and then they'll have to wait for the reports to come back (in around a month, naturally) to find out the results of their gambits and plan their next move. Given your comment about your players' feelings about all the side quests, the Brevoy civil war has the advantage of feeling like the "main quest" to anyone who has read the campaign's background information.
And of course you could create slow-moving scenarios of your own to help pass the time. Maybe a plot involving Werewolves can only be investigated on the middle night of the full moon, so the players have to keep waiting a month between each step of the investigation. Maybe a fairy ring in the Narlmarches only appears when it rains. Maybe the Old Beldame assures them that trying to win over the goodly fey of the Greenbelt requires leaving offerings on the stone slab on summer nights when the moon is gibbous and waxing. Or you could tie into things happening in the actual kingdom turns - maybe an NPC they're friendly with always wanted to be a tavern keeper, and if they build a tavern then some story begins which requires other buildings or improvements to progress, so they need to "unlock" each step of the story bit by bit.