Royal Leamington Spa
South towards London, West or North towards Birmingham
You need to get to the M40. From here you can get to pretty much anywhere. The problem lies in getting to the motorway. The best place is unquestionably the Warwick Service Station, which is between junctions 12 and 13. There is a large station on both sides of the motorway so you can go both ways.
To reach Warwick the Service Station by walking: Starting at Tachbrook Road, follow it down until you reach Harbury Lane. Turn left. Keep on this road until you hit the Fosse Way. Take a left at the junction. Just before you reach the M40 itself, you will see a small road called Windmill Hill Lane. Take a left down this road. After a couple of minutes you will see a turning to the right for special access vehicles only. This is the back route to the services. Follow this road for another few minutes and you're at the Warwick Service Station. From here it's really easy to get a ride to Oxford or London. The walk does take a while though (it took Theinefficienthitchhiker 1h30). It's about 6 miles from Leamington Centre to the services. There is a place to walk alongside the road (grassy edge) most of the way so it's not that bad.
Follow the same route as described above. However, do not turn down Windmill Hill Lane. Continue along the Fosse Way over the M40. Take the immediate left after crossing the bridge, just before the roundabout. Follow this road for 5/10 minutes and you'll reach the Warwick Service Station. From here you should be able to get a lift heading towards Birmingham or further.