A holding deposit of up to 1 week is normal when reserving a rental property. It is governed by the Tenant Fees Act 2019. The deposit is forfeit if you pull out without good reason; if the tenancy goes ahead, then it must either be returned or deducted from the first month's rent; if the landlord pulls out, then it must be returned.
This does sound a bit dodgy. Reputable letting agents should not be having problems relating to their accounts. Are they registered with a professional trade body (e.g ARLA) and an official redress scheme? Which official deposit scheme will they be using if the tenancy goes ahead?
It is possible that they are genuine but a bit rough around the edges...but it's also possible that they are not.
One thing you could do is to contact the landlord directly. If the letting agents won't give you the contact details, for a small fee the Land Registry can provide you with the title document for the property, which should include the landlord's details.
Failing that, if the agents are unable to convince you that they are genuine, walking away may be the only solution.