Depending where in the Midlands you live, do remember that a reward flight saver also gives you a free connecting flight, so you need to check that it's not cheaper to get to Manchester or Leeds first.
If you're near the Chiltern Line, one option would be to try and get an advance fare to Northolt Park station, then take the 140 (I think) bus to the Central Terminal Area, then head over to T5 from there. Yes it would take a while, but you say you are time rich.
There's also a bus from High Wycombe but this will cost more than from Northolt. Not sure of exact fares so you need to do the analysis.
Using a similar line of thought, depending on where you start an advance fare on FGW might get you to Slough where you can hop on a local bus to the airport.
But that said, megabus into London and tube out is probably going to be cheapest overall.