chrisc79:
Sorry I don't know the exact answer to your question but yes, you are right the switching process is managed by the gaining company and the losing company's final reading(s) and the gaining company's initial reading(s) must be exactly the same and on the same date, so that is the first thing to check.
Not sure that would exclude the possibility of BG scr*wing up their billing (indeed they have a reputation for so doing) and potentially back billing for other reasons. AFAIK there is no prohibition on a genuine mistake being back billed but only up to 12 months and there can never be different readings on the two bills.
An unsolicited phone call is a totally unacceptable means of contact. You should insist on a proper bill and contact by mail not phone. I would also recommend that you phone BG to agree a documentary process and timeframe, not a payment, and if BG fail to deliver on the process and timeframe complain scrupulously in accordance with the BG complaints process.