On completion day though do I have to have my whole house packed up and waiting in a van - I can't move in stages over a couple of hours - it's literally instantaneous when completion happens?
Yes.
You don't own the house you are buying until the money is transferred.
You, presumably, can't transfer the money, until the person who is buying your house has transferred the money to you (well, your bank, and your solicitor is happy). Potentially, the people buying your house, might be waiting for the money from their buyers, before they can transfer to you.
So you have to take everything out of your house and on to the van in one journey normally - I've never moved into a new build to know if they hve any leeway
How much does a removal van cost for a 3 bed semi?
Sorry, I can't help with that one - it's a long time since I have moved, but also it will depend on how far you are moving, which day of the week, how much stuff you have, if there are any issues with access, etc.
That said, removal men are incredible. Seriously, they can pack up a house in the fraction of the time it would take you and an army of helpers. Unless you are in the situation of moving from rented, or from parents' house, and can do several journeys over a few days, I wouldn't try and scrimp on a removal company.
Or can I hire a van big enough myself? But I'm guessing I'd need someone with a licence able to drive it?
I can't see you getting the contents of a 3 bed house into a van small enough that you don't need a special licence to drive.