If the property you are buying is empty and you don’t not require a mortgage, the transaction can in theory be completed in a few days but this is highly unusual.
The more likely scenario is that there will be a chain of transactions and a mortgage will have to be arranged. It such circumstances it will usually take 4-6 weeks to exchange contracts and then another 2-4 weeks between exchange of contracts and completion making a total of 6-10 weeks in total.
