What's your mum's opinion about where she lives? This is crucial! As she's self-funding, you can move her to a care home yourself, if she agrees to it. Maybe you could wrap it up as a few weeks convalescence? Or tell her the boiler needs fixing so it's just till that's done. (You don't need POA for health to do this.)
Getting her there is the first thing, and you've found a couple of places so you could just go ahead - you don't need permission from SS. However, once there, it depends what happens next. If your mum is happy to stay there that's all good. If she decided she wanted to go home and got vocal about it or tried to escape, the home would have to get a DoLS to keep her there. This is s Deprivation of Liberty Safeguarding Order and is a standard requirement in locked care homes. It gives them permission to keep someone there, effectively against their will but for their own safety. If this happened, good old SS would become involved again. They would assess your mum's capacity to decide if she was able to understand her care needs. If she doesn't and the DoLS is granted, all well and good. But... if she insists she wants to go home and SS deem her to have capacity to decide, they will then allow her home with a care package and you'll be back to where you are now. i.e. waiting for something really bad to happen that proves to SS she needs a care home.
If you wanted to try this now, you could wait a few days (till the OT has gone...) then do it. But as I say, much depends on what your mum wants (or says she wants.)