I recently replaced my bottom bracket and it seems to work fine. However, since I did this I've had a problem with the front mech - basically the chain won't lift onto the largest of the front chain rings when I attempt to shift up to the higher gears. The rear mech is fine but basically the front just stays on the middle ring; the thumb shifter just feels totally slack. I did also notice that when I shift down to a low gear I have to push the mech in with my foot before it drops in. This suggests to me that the front mech is possible seized up in some way? Should I take it off and attempt to clean it? Or spray a load of lube and let it soak in?
Return of the Mech
Detach the cable from the derailleur and try sliding the cable back and forth through the housing as well as trying to move the derailleur back and forth by hand. Probably one of the two (cable or derailleur) is binding. A lot easier to fix once you know which.
I wouldn't take the derailleur off the bike yet. The grime on the exterior of it won't usually cause a problem like this anyway. Wipe it down a bit and drip some chain lube on the joints. But moving it by hand while you watch closely what is happening is your best bet to spot the problem.
Does the new BB have the same axle length as the old one?
Your symptoms would suggest it does not, and you need to re-adjust your front derailleur (FD) to match; which will involve disconnecting the cable, adjustment of H and L limit screws, and possibly vertical movement of the whole FD on the seat tube.