Still, I think the choice of RR or P&W is based on financial considerations for the most part. Two examples come to mind:
1) AA. The airline chose the -535 for their 757's when the order was anounced in 1988, AA had chosen the F100, with RR Tay's. AA probably was able to negotiate lower prices per engine with RR, for purchasing a much larger amount of engines in total. It was a smart move. In 1981, AA had announced options for 757's with PW2037's, but did not order the planes.
2) UPS. They ordered 30 757PF's with PW2040 engines. Their follow-on order for 757's had RR -535. UPS, however, was also purchasing RR Tay's for their 727 re-engining. Again, a lower price to UPS for purchasing a larger total of engines? UPS must get the best deal, because they don't seem purchase only for engine type commonality (767 freighters with CF6-80, A300-600 with PW4158, RR and PW 757's).
As for BA, I'm sure the RB211-535 was a deciding factor in their launch order of the 757 in 1978. Their predecesor, BOAC, ordered their earliest 707's in the late 1950's because Boeing decided to offer the RR Conway on the 707, which the UK govt. might have insisted on? The later 707-320 ordered by BOAC had JT3D-7 engines. I, too, live in Connecticut, and hope that P&W does well, but they have to compete.