I have dealt with HSBC, LLoyds, Santander and Barclays regarding the POAs I currently deal with.
None of them have restricted online access and I don't see any reason why they should. They should all add a login for the attorney to give them full access and leave the donor's access as is.
HOWEVER, I will add a note of caution and have a little moan at the same time...
We found a dormant account my mother had at LLoyds but even though I have POA, they would not reactivate it unless my mother visited a branch personally, which she is not able to do. Luckily, there was only a very small amount of money in it so it wasn't a big issue and I decided not to make a fuss about it. If there had been a large amount of money in it, I don't know what I could have done.
We also found a dormant account at HSBC and they took the dormant flag off it no problem and gave me full access when I registered the LPA with them.
Santander won't issue a replacement debit card on my aunt's account unless she goes to collect it from her local branch, which she cannot do. They also will not issue a debit card to me, despite being registered as the attorney for her with them. Debit card issues with POAs have been highlighted before with Santander so they still don't seem to have sorted it yet.
The bottom line is that all the banks make up their own rules with regard to POA registration and access, so it is best to ask them what their registration procedure is and what access attorney(s) and the donor will have BEFORE registering the POA with them so you don't have any nasty surprises later.