If there is a chain involved - forget about it being any quicker than 6-8 weeks - it's usually this amount often for just buying a property or selling a property with no chain.
A purchase of a flat, recently for a friend of mine, was completed in 4 weeks - but that was a plain purchase from a builder with aevrything in place. If there are valuations, mortgage companies two firms of solicitors - searches (which can be done on line) you are talking realistically6-8 weeks.