The only non-stop flight from TPA to London is BA. They are in the AA alliance. AA is probably the choice for your routes. There may be some others that will servvice parts of the Caribbean non-stop from TPA.

Depending on where you are going to live, MCO may be an option every so often of the flights are non-stop. Tampa Bay is big so where you live will make a difference to using TPA, PIE, and MCO. In a pinch there is always Sarasota but the airport os pretty small and limited like PIE.