Even if the purchase is buy to let, the property must be vacant on the day of completion, unless the mortgage allows for tenant in situ, which is unlikely.
If I was in your situation, I would ask for a AST starting 1 day after completion. You could leave your stuff in the flat (no valuables), I would advise you to have a van to store your goods just for 24hrs.
If you completed on the sale at 3PM, you would not be able to enter the property until after midnight which is classified as the following day, as the new owner might want to change the locks, check out his property etc.
You have to be exact with the time period, normally an AST is for a min of 6 months, approx 5 months as a timescale would not be acceptable by me as a I would need an exact date when you are vacating. You may also have to allow the buyer to have prospective tenants visiting the place after 3/4 months.
I think its something you can remedy with the buyer, you have spoken to him, ask him /her for a contract 1 day after, you need that asap, as word of mouth will not suffice. Consulting a solicitor is a last resort