"I have sat my laptop next to the PC and connected to O2 with no problem and at a reasonable speed."
If this was done wirelessly, then the PCI Wi-Fi card in the desktop may be at fault, in that it has limited range.
Try to tweak its antenna - ensure that its screwed in fully & move it about. If there's no change, then try a USB Wi-Fi adapter if you have one / can borrow one. G