Yeah, booking deposit ASAP then survey soon after.
Your times are off though. The more delays you introduce the longer it will take. If you want to speed up the process I'd suggest getting the survey done sooner than 1-2 weeks.
No two transactions will be the same as the there are so many disparate factors.
The minimum would be four weeks but a complicated sale can take over a year.
My current experience may guide you:
I paid a booking deposit four weeks ago. As of last week the solicitors had not exchanged contracts. My solicitor said we'd sign the contacts "about three weeks" after she received them. Then we could expect to her the keys "about two weeks" after that.
Based on this, I'd expect to to get the keys the last week of next month.
Note, these times are based on no hiccups.
They are also based on no bank involvement, i.e. we're cash buyers and the vendors own the house outright.
Edit: note, I won't be cracking open the Champagne 'til we get the keys; we may have some Cava or Prosecco when we sign contracts.