On completion the buyer's lender releases the mortgage advance, the buyer's conveyancer sends the balance to the seller's conveyancer, and once received the seller's side authorises release of the keys. The transfer deed takes effect and the seller must vacate.
After completion the buyer's conveyancer has 14 days to file and pay SDLT and then applies to register the new owner (and any mortgage) at HM Land Registry. Registration itself can take weeks or months but ownership is effective from the completion date.
Completion dates are set at exchange. Missing completion triggers "notice to complete" (usually 10 working days) and daily interest; persistent default can unravel the contract and forfeit the deposit.