The rear wiper is NOT part of the MOT.
Even if the wiper isn't working, the wash should work - the pump is in the screenwash tank at the front.
I recently bought a new 3rd party rear wiper from ebay for about £50, works fine.
The hardest part of the job was getting the old wiper arm off - I ended up buying a wiper slide hammer kit, like this:
https://www.ebay.co.uk/itm/S-X8WA-8p...item33efe08a92
That got it off in seconds!