I would just rent a van and move them myself.
When you’re selling and buying on the same day, it’s not really an option. Removals generally arrive at 5-6am so getting a van would be quite a task. Also; a self store unit for a month is around the £200 which is roughly the cost of just a weekends van hire.
There’s a boat load going on, once your old house is empty you need to do the final sweep through with the vac, mop and take pictures all whilst waiting for a solicitor to call to confirm you now own the new house and, you must be ready to immediately vacate the old house because it’s no longer yours (also needs to be in a suitable condition).
Then it’s throw everything in the car and get to the new house, preferably before the removal men arrive with your possessions so you can direct them into each room.
If you’re lucky, you have time for an evening meal but, forget breakfast and lunch… and this is using a company. The people who sold my current house used a “mate” with a van; it took them 3 hours longer than it should to vacate simply because they’d penny pinched. If I wanted to, I could’ve gone mad but it wouldn’t have achieved anything.