But the question is where do they discharge her to? If they determine she has no capacity then they can't send her home alone even if that's what she wants and without a POA you can't tell them to. Even with a POA if the hospital held a best interests meeting they could override your POA anyway.
My mother ended up in A&E after passing out a couple of times, the hospital said they wouldn't let her go home alone, however, if I took her home to live with me then that was OK, I don't remember being asked if I had a POA but I didn't and it wasn't a problem.
If I'd told them to send her home alone then maybe they would have told me the same as they told you, they have a duty of care to her and can't sent her somewhere if they believe she may be at risk so they really don't have a choice but to keep her as a "bed blocker" so it's as much in the NHS's interest to get the assessment done as soon as possible as it is in yours and hers.
The NHS won't want her tying up a bed for the next 6 weeks but if they can't do better than 6 weeks to get an assessment done then they won't be liking it either.
Depending on how the POA is worded could you use an Independent Mental Capacity Advisor (IMCA), some are doctors if that's the way the POA is worded others aren't but they're the people who do hold a professional qualification to assess mental capacity unlike social workers and solicitors who are allowed to assess capacity by virtue of their professional qualifications not a specific ability to assess capacity.
If it's worded that it has to be a doctor or a psych then it's a problem but if says a "qualified person" then an IMCA might be an option.
Usually it's solicitors who put clauses in like this and to be fair it does give the donor of the LPA a level of protection from unscrupulous relative putting someone in a home and pillaging their wealth, not that I'm implying that's what your doing in any way, but difficult for you and costly for the NHS you can see why some people are advised to do this as a safeguard.
I don't see why the hospital can't section her if they can't safely let her out, that way they could have a best interest meeting and put her into care with or without you having an LPA.
A best interests meeting can do this against the LPA donor's and holder's wishes if they want to it seems odd that the NHS are happy to keep her for 6 weeks at great cost instead of either sectioning or getting one of their psychs up there a bit quicker.
If you want her to be sent back home and the hospital and social services don't, then even if, or when you get the POA activated they can still do what they think is in her best interests and over ride any input you have as a POA anyway.
It sounds like you're dealing with this from a distance which to be honest just makes it harder for both you and the NHS/SS, they're the ones on the ground having to deal with it not you, sorry if I've got that wrong but dealing with people remotely isn't the NHS/SS way of doing things so having or not having a POA may well not help you as much as you may think, more than likely it's just a tactic they're using.
Sorry to sound a bit negative but I feel the decisions will be made by them and they're just stalling for time to consider the position then tell you what's going to happen.
K