I'd say you'd want to take the bus from Heathrow to Oxford (2 most hours, 3 per hour at peak times, service all night, free wifi and power points!), and consider spending a bit of time there - it's a great city. Then, go to the station and buy a Cotswolds Line railcard. Now, make your way along the Cotswolds line from Oxford to Worcester, stopping off as you like. Your railcard will get you 1/3 off the price, so it shouldn't cost too much to buy lots of single tickets between stations along the route, and you can just buy them from the station - no need to get them in advance. I'd suggest Charlbury, Moreton in Marsh (+bus diversation as someone else detailed), and maybe somewhere else. Then, see a bit of Worcester, it's a pretty city too. Finally, direct train from Worcester to Birmingham, one change at New Street, then onwards for 10 minutes to Birmingham International for the airport.

The Cotswolds Line Railcard is great for journeys between Oxford and Worcester, but no-one outside of the area knows about it, and it certainly isn't marketed to tourists. You can buy it at Oxford station, just ask at the counter and they'll dig out the form (usually photocopied!)