BA/Iberia flights leave quite often from LHR to MAD. The advantage of a BA flight is that you are in T5, however if you miss it they will likely put you onto an Iberia flight which is from T3 and your baggage has a good chance of going walkies too. Being on Iberia avoids you having to change terminals in Madrid, but then you would have to do that at Heathrow which is just as bad (if not worse).

Taking a Club Class trip and having to drag my luggage from LCY to LHR doesn't make much sense to me, the LCY trip is ok but nothing to write home about. If you are going to do it, then the tube is much cheaper and is often just as quick due to having to wait for the train at Paddington. There is usually an LHR flight at the same time as the LCY flight anyway, so why don't you just take that one, perhaps availability...

Where are you headed to in the South of Spain?