To make sure if the GTX 560 was the problem, try it on another sufficient system, and see if it's like that too, if no then something must be wrong with your system parts (great idea to check them one by one), but if yes it's like that, before getting a 6870, try RMA-ing. If not successful then get the 6870, it should be fine.
I have tried running the system with just one memory stick and both worked individually.
I dont have any friends that have computers (they all have macbooks or consoles!) so I dont have another system that I could test the GPU on.
The computer runs fine for everything else - web browsing etc, it was only when I tried to launch a 3d program. I left it to transfer files across a network and after 3 hours of transfer the system temperatures had barely changed.
I will be getting the 6870 in a few days, hopefully it will solve the problem.
Do you think I should just wipe the hdd and start again to prevent driver conflicts or will this not be an issue since it will be a radeon?
Don't bother with the hwcompare link above OP, this site is not accurate, and this is a comparison of a 6870 vs 560 Ti. In reality, the 6870 should perform the same as 560 (non-Ti). Yes, it'll be a decent Gaming rig, but I wish you can get something better like 560 ti or 7850.