If a single PNR the bags should easily be tagged all the way through. If separate PNRs I believe it should be possible but it will be a bit slower/harder so I'd definitely leave extra time.

For BPs even if they do manage to print them all out in RDU (and I'd bet against that actually happening) you'll want to get "home" BPs from the operating carriers along the way and confirm that the interlined bags are correctly reflected in the operating airlines' systems. The place where it will probably make the most difference is with the transfer buses in LHR as I've oft been asked for proof of the onward flight to be able to use them.