Had both front wheel bearing assemblies replaced recently

Wheel bearing? If you turn left and it gets louder. It is the right wheel bearing. And vice versa.

to possible answers
the axles are new and the cv joints mat be firm resulting in noise transmission through failing engine mounts ( unlikely but possible )
the tread pattern may be the cause of the noise especially if the tyre pressure is incorrect
Take it back to the shop and take them for a road test to see if they can find the reason
when drive axles have to replaced , it generally means that the front lower arms have to be disconnected so a wheel alignment may also be in order

sounds like a right front wheel bearing. when you turn to the left it puts more load on the right bearing. when you turn to the right it puts more load on the left bearing. These are a bolt in style bearing and they are fairly easy to replace.

If you think you would like to give it a try let me know and I'll get you some instructions.

I had a similar issue and it ended up being a broken wheel bearing. I even ended up having two, one was broken on my front passenger side, and one broken on my rear driver side. The part costs about $280.00, but could cost around $400.00 with service repair. The noise of a broken wheel bearing can sound like a humming, and over time it becomes louder. The car can be driven with a broken wheel bearing, but doing so can eventually lead to other wheel/tire maintenance. MINI's endure this commonly due to their low profile to the ground. Here is a forum on further issues experienced, including the bearing, and efforts towards recalls. http://newmini.wordpress.com/2009/02/17/mini-cooper-power-steering-failure/#comment-884

check that, and you may have a bad wheel bearing on that side.

it can be any of the above, rotate the tires and see if the noise location seems to change, but the last thing i would suspect is the CV, they just click when turning, the wheel bearing is my choice here, they roar, tires make a diff noise.

caster,camber and, wheel bearings
have your car 4 wheel aligned this will adjust your rear caster and, camber.
have your wheel bearings checked

The noise you're hearing is consistent with a bad wheel bearing. It's true, the only way to actually tell is to pull the hub/bearing. I'd try to isolate what side of the car the noise is coming from and pull that wheel. If it's not a wheel bearing, it could be transmission related but these things are usually solved by replacing the wheel bearing.

Sounds like a wheel bearing going, or gone. Possible tire/alignment, but more than likely, wheel bearing. You going to fix, or have it fixed?

It really sounds like a wheel bearing to me, If you drive down a straight smooth road and gently rock the steering wheel back and forth, you can sometimes tell which side it is by the way the car loads the front wheels. I believe this bearing can be purchased as hub and bearing assembly so it is fairly easy to replace but the part is pricey. Remove the wheel, brake caliper and rotor, look through the hub flange access holes and undo the mounting bolts and pull the bearing assembly out of the knuckle.