Any suggestions?

Poor eyesight and high bp.
I’m supposing these are being treated in which case please don’t worry too much.
She’s probably most comfortable making her way around her own home with her vision and providing she’s taking her meds for her bp why should anything go wrong if your not there.
Unless she wants to cooperate you must try and see it from her point of view and perhaps contact her gp to see if maybe she could have an occupational therapy assessment whereby any additional household aides could be of benefit if you think she maybe likely to fall.

ExD, there are variations on how you can wear a personal alarm - some can be incorporated into a key ring, (could be kept in a pocket?), pretty bracelet, watch-type strap, etc. Even a small and pretty "fob", which can be "pinned" on, like a brooch. Overall I agree with NemosMum; you can only do your best. In your place, I'd simply let Mother get on with it and take the consequences. Sounds harsh, I know, but IMO you can't help someone who refuses to be helped.

thing is, while i understand the concern, many of these suggestions are not relevant in this situation.
the person has the absolute right to live as she wants.
she has no legal obligation to live in a way that her relatives approve of. or that they deem more sensible.

When my FiL died in November, one of his sons installed CCTV cameras in the house so that they could keep an eye on MiL when she was alone. (Her unmarried daughter lived at home but worked. ) She had had a lot of bad falls, so the family were worried about that, although she had one of those pendant alarms too. She also had a carer come in for part of the time, so I expect that BiL was spying on her occasionally to make sure that she wasn't mistreating his mum. I don't know for sure that he did but I expect so.

MiL died in March, so the system wasn't in use for long, but my BiL said that it was good to be able to check that she was OK during SiL's working hours. I thought it was a bit intrusive but I knew it was just because they were worried about her. I suppose your mum wouldn't go for the idea?

Getting a social worker to suggest an alarm to you mum is a really good idea.

You are her daughter and as such she will pooh-pooh any suggestions you make.

Is she one of the blunt people who can't stand anyone talking bluntly to her? If not, plain speaking, telling her how worried you are might help.

I would do it anyway, whether she likes it or not. It will ease your conscience even if it doesn't do any other manner of good.

Nemosmum and welbeck read my mind

You’re mum has to be wanting and accepting of help
I totally agree with welbeck if your mum has complete mental ability you have to leave her to ask for help and be prepared that she won’t ask
I understand how distressing this is for you but until she feels she needs protecting you can’t be bringing people in to do this that and the other if she doesn’t want alarms round her neck or Sensors watching her every movement there’s nothing you can do apart from spell it out to her that if she falls or has an accident she may lay there for a couple of days if she is knowing and accepting of her own fate ...that’s it You can do no more

Blimey the thought of ‘watching’ her sounds horrendous bluecat and a complete Invasion of privacy unless the elderly person asks for that to be done
Makes me think of Geoff spying on Yasmine in Corrie
Different if the old person asks for it if they think they are being abused or a carer is thieving but otherwise horrid

Leave her to it! It’s better than having someone who constantly needs you and whinges all the time. She’ll let you know soon enough. There’s no more reason to put up with this from an older person, than you would if she were younger. Enjoy time to yourself.

Please push for Power of Attorney, both legal & medical. It's such piece of mind having it

Blueberry has the answer. That way your mum will still feel in control.
My MIL was the complete opposite. She would phone my DH up to change the tv channel because she was so impatient & would just stab buttons - any buttons. Then we found a simplified control with extra large buttons. She'd hide her hearing aid etc. There was a resident Warden but wouldn't use her. We can smile about her now but it wasn't funny at the time.

Bluebelle, Yes, I was in two minds about it myself. However, DH's family are a law unto themselves and it is useless to argue with them. BiL just decided to do it, and went ahead.

I did think it was quite funny when BiL eavesdropped, after MiL's death, on his three sisters bitching about another member of the family. He told them off, rather sanctimoniously, and they told him in no uncertain terms what he could do with his equipment!

It's now installed in a widowed aunt's house, at the request of a cousin.

Thank you, ladies, for your help and support!
She is fine mentally- very keen on news and politics and extremely sharp on UK/world affairs - no mental problems at all!
We have a reasonable relationship- not close but ok. She has always been awkward - very prickly and inclined to take offence and once offended, the insult is never forgiven. Siblings were kept at arms length and never seemed important. She is adamant that there is NO need for Power of Attorney.
I should have mentioned that, after 18 months of nagging, she did agree to an alarm, was rude to the person who installed it because "it was an intrusion" and then refused to use it.
Please understand - I admire her as do my children & friends, but it would help if she could see that it makes things difficult for me. Her answer is "What will be, will be." - no help to me, trying to work out whether to call 999 or not!
I posted this thread because there must be people out there who have mothers like this. The ones who fought to be seen as capable, & clever, and who, having got their independence, don't want to lose it. The problem is - how to deal with them when they're deteriorating?
Thanks for listening.

it sounds to me that maybe she does not want anyone to call 999, ever ?
i can understand that.

My mum is difficult too. We had a personal alarm for round her neck but she fell the other night and spent twelve hours on the floor before pressing it. Her answer was I forgot about it. She won’t take any personal responsibility for anything. She is immobile as she hasn’t moved for the last forty years and her muscles are weak but she refuses to accept this, just saying it is old age. She also refuses to leave her huge house, despite living in only two rooms now. She would be impossible to live with as she hoards things and won’t go to bed until three in the morning! My sister and visit just once a week as she won’t move to be nearer either of us. We have unfortunately just resigned ourselves to the fact that she will have another fall before long and when she breaks something she will have to go into a home.

It is so frustrating as she could choose an apartment that had everything to make her life easy and at this point choose her surroundings. However it is her life and she must make her own decisions.

Maybe do a Covid postal test and when the results come back go and visit her. As said previously the aggressive attitude to people could be onset of dementia and it would be good to get some safety plan in place. Good luck, I feel for you.