Tell us the real deadline
Say when you need the data and which files you need first. If a few folders would solve the immediate problem, we can copy those first and return the rest afterwards.
Save a day in transit
If you are in or near London, you can bring the device to the lab in Wimbledon yourself. Call first so an engineer is ready for it. From elsewhere in the UK, a next-day courier is the quickest route.
What a faster service changes
Express and Emergency move your job to the front of the queue. They cannot make a damaged drive read any faster, which is why the times are targets rather than promises.