Do I need a visa if I'm staying in Thailand for 31 days?

You will probably be ok. One concern is the overstay and any future business you may have with Thai immigration. If you plan to visit only once in your life then you're fine. If you think their maybe a possibility of return trips I would try for the visa or get an extension while your are here (visa is better option).

Are you going by the date you leave the uk or the date you actually arrive in Thailand to make it 31 days?

It’s not Visa on Arrival (which is 15 days for specific nationalities) It’s called visa exempt- you don’t get a visa at all.

That being said you’ll probably get away with being one day over. Usually they won’t fine you for one day, but they could. It’s 500 baht per day overstay.

The hang up you could have is your airline not allowing you to board as you don’t have a flight out within the number of days.

The cost of a tourist visa is approx £30. Probably best to get one and save yourself any hassles at either the check in desk or immigration on arrival.

You’ll save a third of that cost by not having to pay overstay fine (if they don’t waive it).

For a one day overstay they will waive the 500THB overstay.