But this is where I can't understand who is advising you or what is going on @roundaboutthehillsareshining. Tjese things take time, as do all the assessments. Even doing the assessments now is risky, as in a few weeks at home your relative's condition may improve (ie no hospital delirium) and they may be found to have more capacity than any one thinks. However you have now listed some specific queries, so I will do my best to answer them.
As I said I have been through this recently. You can't apply for Deputyship without a separate MCA for finances and, they will most likely turn it down without a diagnosis. To get a diagnosis we had to push for a D2A place otherwise we were back in the while GP/memory clinic doom loop which takes months unless there is a further deterioration and admission to hospital.
When we were at the stage of applying for Deputyship it took 3 months from admission to hospital to work through all the paperwork for Deputyship for DM. It was then about 4 months for emergency Deputyship to be awarded. You either need to make alternative provisions for her or let the system try, fail and pick it up again in the meanwhile.
However (and this is a big however), smeone who is deemed to have capacity - however limited - is allowed to make their own decisions. And you may find your relative hasn't got capacity for decsisions around her own care but does over financial decisions, or certain types of financial decision. The reason these are separated out legally is so that it reduces the risk of abuse. This is assessed at the Care Act Assessment, the Deputyship application and again during the DoLS process on admission to a residential setting. It will be reviewed via DoLS regularly (or should be).
In the case of my DM we were awarded partial Deputyship because she is deemed to be able to decisions about certain things. And then, and only then, if I involve her in all decisions. Yes it is a nightmare. Is there anything I can do about it, no. Her MCA found she was able to articulate her wants and wishes when it came to managing her finances, even though she can't access her finances without support.
In the meanwhile, SS are legally bound to try the least restricive method of care first. This means supporting your relative to remain at home, until such time they have enough evidence to show she needs additional support. Am sorry this isn't what you want to hear, but they will discharge with no access to funding and apply for food bank support if no one from the family is willing to pick it up. We couldn't pick it up in DMs case. Running two households wasn't feasible at short notice. DM was back in hospital within a week having refused carer support. It was awful but it was the only way the system would pick her up, by then her care needs were too great for us to support. The system is horrendous.
In terms of the Deputyship in all, it has cost us in the region of £6K over a 9 month period to sort it all out. (DM is now in a care home but only after a very prolonged stay and deterioration in hospital). If DM had been at home I would have expected double that just on household management. However formal care costs are "loaned" by the LA until such time as the Deputyship is in place.
As long as you keep all your receipts all funds can be claimed as expenses under the Deputyship (including those associated with the application) but it is a lengthy process to put it all in place and manage the reporting etc. If no one in the family wants to / can afford to apply, the LA will apply instead and appoint a solicitor to oversee the Deputyship on behalf of your relative. Why won't immediate family pick this up? It is unusual for extended family to apply. And you will have to get support from the family to apply for Deputyship which could delay the application. If you don't, it will almost certainly be turned down.
My final advice is phone Dementia UK helpline or try their specialist forum. And if you can afford it, use a solicitor to do the application - it takes some of the sting out the nuclear button.