Can I get my tyres changed during an MOT and Interim service?

Usually. Just ask the garage to let you know first. They do the service and repair work first, before the MoT. You could, of course, go to a tyre place first and get their opinion.

Whatever anyone answers will be a lesson for me.

Your MoT tester is not supposed to work on your car during the roadworthiness test because it creates an obvious conflict of interest where they tell you your car needs this and that in order to pass the test. This would be open to abuse because your natural instinct would be to agree to the fitment of certainly overpriced and probably unnecessary parts. If you fail the MoT I believe you have a grace period where you can get repairs done and retest for no charge. Also, most garages don't change tyres; they just send them out to a tyre place they have a deal witj and mark up the cost to you. So if you fail because of your tyres, go to a tyre shop yourself and have them replaced there. Tyre shops have most common sizes on the shelf and can fit you in quite easily. Don't make the garage who does your testing do any maintenance/repairs unless you are a regular and trust them.

If you leave instructions to replace the tyres if necessary.

You also need to leave clear instructions about what to do if there is a significant failure. You don't want to spend money on a service( or tyres) if the subframe is rotted through.

Leave a number where you can be contacted and answer all calls.

If you've got the cash they will sort the tyres. If they need changing they will need changing in pairs, either fronts or rears.