You picked the wrong half of Oxbridge to stay in!
Oxford has two buses an hour to Heathrow for virtually the whole day, and buses overnight too. The buses have wifi, powerpoints, nice seats etc. Journey time from the city centre to T5 is a little over an hour if traffic is fine, or about 90 minutes to the central bus station (for T1-T3). A couple of weeks ago, I managed to do Oxford Gloucester Green Bus Station to the BA Galleries South Lounge in T5 (including bag drop and security) in just under 80 minutes, so it can be pretty quick! Traffic today wasn't so great, so it took 1 hour 45 bus station to lounge. Still not too bad :)
If you get the bus / train / underground to T5, then you're virtually in the terminal. It's normally 2-5 minutes until you're at the departures level, just needs a little bit of walking and lifts/escalators up (departures is on the top floor). With T1, T2 and T3, all the public transport heads to a central point in the middle of the three terminals (either the central bus station, or something just under it). If you run, you can probably make the terminals in 5 minutes. At a slow walk, using the lifts and travelators, it's about 15. If you take a look at a map you'll see why! Don't worry about the transfer though, just allow time.
(T4 is somewhat of an ugly duckling these days, transport links wise. Allow yourself lots of extra time, as unless you're very lucky you'll need to change bus / tube / train to get there)
Assuming you can't be tempted to head west to "the other place", I'd suggest you look at south west London. You ought to be able to find somewhere with a direct bus link to Heathrow, that isn't too pricey, and hopefully with some nice pubs and parks around. Getting from Cambridge down that way will need a few changes across London, but at least you'll then be nearby for the early-ish start to the airport. Try looking on the TFL website for bus maps to get an idea of where to go for.