Sorry if this is elsewhere, but seach did not find.

I am looking at Business Club World flights from Phoenix to Madrid and return.

1. On the outbound, they have a connection on 1 hr. at Heathrow at 13:05. Is that enough for me and my bags. (I will likely stay at an airport hotel in MAD for a night - so guess bags could catch up to me.) Also, if I miss the LHR-MAD flight, will B.A. put me on the next one without a quarrel?

2. On return, there is a flight from MAD to LCY (City) that arrives at 09:05. The connection flys out of LHR at 14:35pm Is that enough time? What is best (and most reasonable) way to transfer from LCY to LHR? Do I need to collect bags, haul on the bus, and then re-check in LHR?

Thanks for any help.