You will need to contact DVLA, First to change the tax category to Private HGV, you are right, you cannot legally use a restricted use agricultral machine for private use
if the Unimog can be verified as manufactured 1979, unless it was given a Q plate for other reasons, ie. re built from parts, it may be possible to re register on an age related plate.
Then contact DVSA to arrange for a goods vehicle test as its 6000kg GVW, it does not need seat belts if it can be confirmed as manufactured 1979, belts in trucks were required from 2001. If it has a Tachograph, it may need to be calibrated for a first test, though for private use you are not required to use i
And of course you will need a pre 1999 issued licence to drive it as it is a goods vehicle of 6000kg Gross weight, its too heavy to downrate to under 3500kg GVW.
https://www.gov.uk/annual-test-for-lorries-buses-and-trailers/vehicles-that-must-take-the-annual-test