MOT. Take it to any MOT test centre. They are mostly busineses (not State owned) in every town, usually at workshop garages or similar facilities. Costs about GBP55. Takes about 30-40 minutes. There is no personal identity or other documentation required - anyone can take any vehicle and get it.
It iIs an annual certificate. If the existing certificate has not expired take it along and they will add to the new one any unexpired period of up to a month. If the old one (or the Tax) has expired you are permitted to use the vehicle to go to, and from, a pre-booked MOT and if it fails to go to a pre-booked repair place.

Tax. The vehicle must have a Registered Keeper who need not be the Owner nor the Driver. It can be any person or Limited Company but they must have (or provide) a physical UK address. The vehicle must have valid Tax if on any form of public road (even parked) in UK. If it has no tax it must be declared as off-road (SORN) annually. Its one or the other, tax/SORN.

Insurance. A regular problem for non-UK residents, but not impossible. There are other threads about this. Most use Internet insurance brokers.

You could ask the Seller to tax it forward for a year (and pay him the cost) if still the RK and has insurance for it. That postpones the tax problem for a year.

Are DVLA blocking you being on a non-UK IP address or just not accepting a typed-in enquiry without a physical UK address?
If the former you could try loading "Ex-Pat Shield" which assigns a temporary non-geographic IP address or a UK one (not sure which). It is normally used to be able to view on-line UK TV live, which is blocked to other countries for Rights reasons.

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