Typically yes, EI will give you your BA BP for the ex LHR flight (I've had issues with AA ORK-LHR-USA but never BA). Note, however, if you're in a premium cabin and go landside the LHR ground staff may be confused by your EI issued BP as it won't look like you're eligible for fast track or anything and sometimes the lounge and gate attendants will give you a look.

When I go ORK to connect at LHR and go landside for whatever reason (that reason is invariably a smoke) I have the BA counter print me a new BP to make things a bit easier.