That is a pretty significant hit for all spellcasting. Like massive.
The reason is iteratives - most fighty types get their advantage from ealier/more frequent iterative attacks. Since the a full BAB druid would likely need to wild shape more often, while the higher BAB will kick in, the fact that they have natural attacks vs. iterative attacks means that the benefit of the full BAB will be lessened.
Really, look at the ranger. That's (in a weak way) a full BAB druid, and even the base rules didn't dump all of spellcasting. Is there a ranger AT that would fit what he's looking for?
bab is just accuracy, feats qualifications, iterative attacks, and power attack scaling.
So sure, giving him full bab in place of spells is a downgrade so shouldn't really cause any issues. Instead of being able to cast spells to increase his accuracy and damage he's getting that via BAB and power attack now. Using natural attacks means you miss out on iterative attacks anyways.
So the only bonus the player is getting is qualifying for feats.
Ultimate Wilderness will have a shifter - a full BAB shapechanging class.
Though I suspect the time before it will be released is beyond the time your player is willing to wait.
There's a new class coming out in the fall in Ultimate Wilderness called the Shifter which is pretty much what your player is asking for. Supposedly, it will be a full BAB shapeshifting focused character and has been described as a druidy version of a paladin or cavalier.
If you want something *right now* I'd say giving up all spells for BAB is hefty loss.
I'd up the HD to d10 (like Ranger), increase to full BAB, add a new "Minor Wild Shape" class feature at level 1 that allows them to take on the aspect of an animal and gain a natural attack (a bite, gore, 2 claws, 2 hoof attacks, etc.) as a swift action an unlimited times per day. Make it a polymorph effect, so it can't be used with actual Wild Shape that comes online a little bit later. I might also add something like an Improved and Greater Wild Shape at levels 10 and 16 that reduce the action required for Wild Shape to a move and swift action respectively, and possibly additional uses/day to facilitate changing between multiple forms.
Ultimate answer: Depends on your campaign.
He'd be more versatile than most other full-BAB classes due to wild shape's abilities. Other full BAB classes that have shapeshifting abilities tends to be far more limited - you get claws, or a scent ability. Think "aspect of the [insert animal name here]" spells or feats. They never get things like shifting into plant or elemental creatures either.
From a melee standpoint, he'd be weaker than other full BAB classes until wild shape starts kicking in. Then he'd end up being in barbarian territory at the end of that.
Animal companion would have to get scaled down. Think ranger's version of animal companion.
However, he'd still lose a lot of buffing/debuffing/battlefield control abilities without the spells, and druid spells are pretty good.
Overall, as long as you scale back the animal companion to ranger levels, I think it will work out.
I'd look at the metamorph alchemist. It has a wile shape like ability at first level, and mutagen, and is still 3/4 bab. You get a lot more utility out of animal shapes than you do out of alter self/monstrous physique, but on the other hand, you need wild armor.
It should be okay if you wanted to house rule that, but there could be some unforeseen combos that could make it very powerful. Such as taking an inquisition because he still has nature bond but no spells, taking anger inquisition, and then using rage and furious finish.
With the core druid, it is difficult (at least with a 20 point stat buy and recommended wealth) to excel at casting and wildshape melee. An optimized character becomes a caster with some mobility bonuses from wild shape, or a wild shape melee monster with a few utility spells. Focusing on either tends to hamper the other, I assume this is intended in the class design. So trading away spell casting is not as significant of a penalty is it looks like on paper.