They told me there that it usually takes a few days for apple to authorize the repair after the 1 year vendor warranty, but I was able to pick up my laptop again on the next day in the morning.
The local certified shop wanted £99 to diagnose the problem. Apple weren't particularly helpful when I asked for help to get the fee waived (their staff in the US don't know about the UK Sale of Goods act, understandable perhaps), so I had to take a 40 minute train journey to my nearest Apple store.
The Apple store diagnosed the problem and immediately offered to repair it for free 'under UK consumer law'. I was pretty happy with the experience, but would have preferred if the local certified shop had been able to diagnose and repair the laptop for free instead.