So if the pilots braked immediately, with only the back wheels down, and nose wheel still up, would the force from braking cause the front wheel to come slamming down?
If you get on the brakes hard after main touchdown, once the anti-skid is active following wheel spin-up (you can't land with brakes applied on any airliner with anti-skid - brakes are depressurized until after the main wheels start to spin and/or you are weight-on-wheels for a minimum time) and don't apply any additional compensating elevator, the nose will come down faster than if just hold the existing elevator input, and will contact a bit harder, but "slamming down" would be a bit extreme unless you also relieved some of the control back pressure as you did so. Then it would be a bit firm and if you were reckless enough you might blow the nose tires, and somebody would have a chat with you.
In practice though, on most jets you land the mains, then the nose in two separate actions and during that "second phase" you are applying elevator to modulate the sink rate of the nose to make it touch down reasonably gently for that "2nd landing". If you were on the brakes hard at that point you would normally just compensate instinctively with extra elevator input to get a gentle landing of the nose.
You would certainly have enough tail power available to counteract the nose down torque of the braking at high speed. But you would probably get scolded for getting on the brakes before nose wheel contact anyway, and a lot of pilots will keep their feet low on the pedals to avoid applying brake, then move them up once the nose wheel is down.