For future reference (may be useful to others but a bit late for the OP, I'm afraid), this sort of change to a V5C can be done online. My recent experience with selling a car was good; I notified DVLA of the change of owner online and the new owner had the V5C within 4 days IIRC.
As others have noted, DVLA seems to be having a lot more trouble coping with Covid than many other organizations so the advice is always, 'If you don't have to post something to them, then don't!'
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