I have set up a POA through a solicitor so that, should I want to hand over the management of my affairs to my children, all I have to do is sign the form activating the POA. They would then register it with my bank, building society etc.
I had to have a witness that I was of sound mind when I set the POA up, and I think the same applies to the activation. (The rules have changed since I did it for my mother, precisely because families were taking advantage of elderly relatives. ) If preparations have not been made before the person becomes incompetent - e.g. a stroke - then it has to go before the Court of Protection and it's all more long-winded and expensive.
So, while OP can suggest to her mother that she puts an enduring PoA in place, she can't do it off her own bat. Her mother has to agree. And if she refuses, nothing can be done until a doctor certifies that she is no longer compos mentis and it will have to be done through the CoP.
I agree with Peng, you can't just install cameras in a competent adult's house to spy on her and her friends. I'd be beyond furious if someone did that to me. And how much footage would you have to watch before you picked up some chance comment about getting the best quote from Compare the Market - or whatever else Anne has suggested? Many people, much younger than Op's mother, don't realise how much they are being fleeced by automatic renewals, even from firms like Saga.
I'd be keeping an open mind about Anne's motives at the minute because Op's mother obviously enjoys her company and the lifts, and there is no evidence that Anne is up to no good. I think the best thing to do is to try to get to know Anne better and then make up your mind. She might genuinely be trying to save you trouble by picking up the odd bit of shopping when she lives so close. It's the sort of thing a neighbour or friend would offer to do.
The financial stuff is more of a concern. Op what exactly has Anne done for your mother? Bank details are neither here nor there. We divulge them every time we buy something on line. You can't take money out of an account, only put it in. It's the PIN that's sacred and should never be divulged to anyone. How does your mother get cash? Is she able to get to the bank on her own?