Am I right in thinking the following is legal?
Book MOT at your local MOT test centre, make sure you give them registration number and that it's got no MOT.
Pick up car from wherever it is insure it for the day, drive it back home to test centre, have the test done and then drive it home.
All legal as MOT and road tax aren't required to drive to, or back from an MOT test. You've insured it which is the minimum legal requirement.

There is a bit of sneakiness in that you're driving quite a distance to the test centre but any traffic can confirm you have the appointment that day for a test and you're on your way to it.
https://www.gov.uk/getting-an-mot/after-the-test
It says you can drive your car away to have defects fixed (which you will do yourself or arrange for repairs)
It needs to be in a safe roadworthy condition though, which is your responsibility.