I have come across a few questions about this issue recently.
The VAT registration threshold is currently £85,000. This is the value of taxable supplies made over a rolling 12-month period.
When a person takes over another business as a TOGC (transfer of a going concern) the rule applies slightly differently. In this situation you have to add up the taxable supplies made by both the transferor and the transferee. This is explained in HMRC Notice 700/1, para 3.8 (link below).
When advising a client in this situation do review the TOGC rules and confirm whether or not they apply.