Most people expect this to be awkward and slow. It is one letter from you, and the two firms handle the rest between them.

When you move, your new accountant writes to your old one asking two things. Whether there is any professional reason not to take you on, and for copies of your records. It is a standard courtesy between firms and it is the reason you do not have to have an uncomfortable phone call.
You are not asking permission. You are telling them, politely, through a process both firms already know.
One thing. You sign a short letter authorising your old accountant to release your records and speak to us. Everything after that happens between the two firms.
“People put this off for months expecting a difficult conversation. There is no conversation. There is a letter.”
We ask for the last set of accounts, the tax computations, your trial balance and the payroll records if there are any. Then we set up agent authorisation so we can file for you, reconnect your bank feeds, and check the opening balances actually agree to the last accounts filed.
That last check is the one that matters. It is where we find the things nobody mentioned, and it is easier to sort out in week one than in month eleven.
Just after a year end is cleanest, because the old firm has finished a complete set and there is a natural line to draw. Second best is the start of a sales tax quarter.
That said, do not wait nine months for a tidy date if things are going badly now. We have picked up plenty of clients mid year and mid mess, and it is a fortnight of extra work rather than a reason to stay put.
A clean line at year end saves a week, but it is not worth waiting all year for.
Occasionally a firm drags its feet, usually because there is an unpaid invoice. Your records are yours and you are entitled to them, though a firm can hold on to some of its own working papers until it is paid.
In practice this is rare, and it is resolved with one more letter. If it does happen, we deal with it. You should not have to chase your old accountant for the privilege of leaving.
Bring the question you have been avoiding. You will leave with a fixed price and a plain answer, whether or not you sign up.