As the presenter of a vehicle for MOT it is advisable to make the Tester aware of ANY modifications that you feel may affect the way an MOT is to be carried out.

A welded diff or even a heavily locked diff will certainly require the use of a Decelerometer - which is no hassle or fuss for the MOT station as this is a calibrated and accepted piece of test equipment and is a compulsary item of test equipment to be available to carry out an MOT

There are many types available but this is what they will look like (or similar) usually used for permanent 4 wheel drive vehicles