Marydoll Addenbrooke's has a system called "MyChart", which is very good. I was signed up to it when my breast cancer was diagnosed. The GP told me that as I was now signed up, I would be able to see any blood tests ordered by the GP because they're processed in Addenbrookes' labs. They're sometimes available within hours of the test. I don't think it's an "English" thing, but it is for any patients at Addenbrooke's Hospital (and probably some other hospitals).
Before that, I used to ring the surgery, have a conversation with a (usually grumpy) receptionist, who would agree to get the results printed off and release them to me when a GP had seen and signed them, which used to take 24-48 hours. I used to find the whole palaver ridiculous, but I persisted.
The receptionist usually tried to fob me off by telling me that I would be contacted if any action needed to be taken. Unfortunately, I know for a fact that sometimes they don't. I prefer to keep a record myself and can see trends. If there's an upward trend, I take action myself. I honestly think I save GP time because I look after my own health as well as I can and like to be in control.
Many GPs in England are refusing to give their patients online access to their full medical records (despite NHS instructions), but patients do have a right to see them.