United Kingdom

Cross Country have a Leicester to Birmingham route, and a Birmingham to Gloucester, and I think without Cross Country, you might have to go all the way into London to do it. London to Leicester will be the franchise out of St. Pancras, I think, and Gloucester to Oxford can be done on Great Western with a change at Didcot Parkway, a little south of Oxford.

For Oxford to Heathrow, a bus is probably easiest - the train would probably involve changes, and a bus might even be direct with no stops.