The bank cannot, purely at the customer's behest, unfreeze an account where there is an extant Account Freezing Order made under the Proceeds of Crime Act 2002.
Section 303Z3 requires, among other things:
(3)An account freezing order ceases to have effect at the end of the period specified in the order (which may be varied under section 303Z4) unless it ceases to have effect at an earlier or later time in accordance with the provision made by sections 303Z9(6)(c), 303Z11(2) to (7), 303Z14(6) to (8) and 303Z15.
(4)The period specified by the relevant court for the purposes of subsection (3) (whether when the order is first made or on a variation under section 303Z4) may not exceed the period of 2 years, starting with the day on which the account freezing order is (or was) made.
(5)An account freezing order must provide for notice to be given to persons affected by the order.
The account holder, as a person affected by the order, is notified of the order by the applicant - not the bank so there's no concern over "tipping off".
The holder may then apply to the court to have it set aside under section 303Z4:
(1)The relevant court may at any time vary or set aside an account freezing order on an application made by—
(b)any person affected by the order.
Only then can the bank (lawfully) unfreeze the account.
- Further detail in answer to OP's comment...
An application for an Account Freezing Order is made a under section 303Z1:
(1)This section applies if an enforcement officer has reasonable grounds for suspecting that money held in an account maintained with a elevant financial institution—
(2) ... the enforcement officer may apply to the relevant court for an account freezing order in relation to the account in which the money is held.
(6) [...] "enforcement officer" means—
a) an officer of Revenue and Customs,
(b) a constable1,
(c) an SFO officer, or
(d) an accredited financial investigator who falls within a description specified in an order made for the purposes of this Chapter by the Secretary of State or the Welsh Ministers under section 453;
What channels each different enforcement agency use to give notice to persons affect by the order under s.303Z3(5), above, is a matter for their own internal processes and procedures - but it is not uncommon to hand deliver it to the account holder and email / post it to the bank.
If the account is frozen under an Order but the account holder has not received their statutory notice, then they could consider asking the bank for proof that such an Order is in place and who made the application. This detail should be made available by the bank to justify their freezing of the account, and enable the holder to contact the applicant to arrange service of the Notice.
1 most NCA investigators and financial investigators are designed with the powers of a constable