Thank you both. My biggest concern is that she's in pain. I'd hate to be dragging out the inevitable just for her to pass away anyway (which will 99.9% definitely be the outcome). I'll keep observing her today whilst I do research into the local vets that could possibly help.
Thanks for all your advice, I haven't had a hamster since I was a child, 30-odd years ago, and I'm trying to navigate this without upsetting my children too much in the process as well.
I agree with sourcing an exotic vet first and also sounding out vet practices on the phone. My experience has been that not all vets offer gas sedation before euthanasia (I am not entirely sure if it's law now or just protocol). I now have a vet who says they will only use gas sedation first as they think it's the right thing.
I hear you, worrying if she is in pain, and you could get Metacam (pain relief) from the vet to keep her comfortable. However many vets won't prescribe it without seeing the hamster first, and then you have to upheave the hamster to the vets - which may mean thinking about whether having her pts at the same time might come up anyway. Some might prescribe it - via a telephone consultation.
The staggering sounds very typical of a slowly dying hamster to me. Whether or not she has had a stroke. Yes it is very difficult with children. With our first Syrian I just explained the hamster was getting very old and slowing down and would sleep more and eventually one day he might decide it was his time to go and not wake up - because he was very very old. I explained how he was almost 100 years old in human years. This kind of prepared the way a bit with some adjustment time. I really really did not want to have him pts but in the end it was necessary.
I think she is probably wanting to stay in her nest at the moment - she perked up at the bit of blueberry. But you are probably best gauging that. She may well have had a stroke but that won't be causing pain - just immobility. Some vets will give advice over the phone in these circumstances.
As a guide though, if she is in pain she may have
1) Her ears flat down all the time (not just when she wakes up - it's normal for them to be down when they've just woken up).
2) Sitting hunching
3) A kind of drawn look around her eyes (my vet calls it squinty eyes - as if someone is pulling the skin back above their eyes).
In some ways it's a natural process for them, but if it goes on for some time then yes she may be suffering as unable to go - sometimes they have a strong heart.
I would take one day at a time. Google exotic vets in your region (there may not be many and it may be a bit further away). Phone round them and enquire if they use gas sedation for euthanasia as standard. If they do ask if it's possible to have a phone consultation with a vet.