The whole registration document isn't passed onto the buyer Ash, only a little part of it, which should be signed by the seller. Maybe it's hiding amongst the bits & bobs you say you have?
If not, I would imagine the person would have given you an invoice or at least a signed piece of paper to say that they've sold the vehicle to you. Either way, you can still drive the car if it's taxed, MOT'd (if nec) & insured in your name, whilst you're waiting for the new reg doc.
Hope I'm not teaching mother to suck eggs - just trying to help. Happy motoring.....