@MommaP - You have my sympathy as this is an awful situation and you do not even have the benefit of time to process all that needs to be done and time to organise or sort through a lifetime of your parent's possessions. Whether their home is local authority or privately rented, I would not consider leaving their stuff for the landlord to deal with as this will almost certainly be more expensive than doing it yourself and/or renting short-term storage. If, as you say, much of the stuff is dirty, unhygienic and smelly, any landlord will charge a small fortune or call a 'hoarder specialist' cleaning firm - also pricey.
I think the suggestion others have made about just throwing the real rubbish and boxing other stuff on the condition it is sorted quickly is a good one. One of the problems hoarders have, apart from the obvious not wanting to get rid of things, is they find it hard to make decisions, each item is milled over even if it is dirty, broken, useless and they find it almost impossible to make quick decisions. I would ask your mum if there are a few dozen items she really wants and knows how to locate them! On the plus side, hoarders can rarely remember everything they have so, after a while, she may fail to notice missing 'stuff'. Soft toys etc may be sentimental but hardly essential.
Do you know if it is more likely to just be 'tat' or 'collectibles/antiques' etc? If she had things of value there was no room for you might be able to persuade her to sell items and buy one new nice thing for her new place with the proceeds of the sale(s). If this proved a successful tactic it is one that may be repeated so she ends up with some nice, appropriate bits for the new flat and each will have a story attached to it she can tell new friends/neighbours...e.g. I got this nice cupboard/unit/display cabinet by selling a dinner service, 2 rings I no longer wear and a dozen first edition books....it might give her some motivation and make parting with things easier. Obviously this would be more straightforward if her 'stuff' was collectibles or desirable easy-to-sell stuff. If she no longer has a record player but has a collection of vinyle records a shop or collector might take the lot...if you suspect there may be a few gems amongst her possessions put them into one of the storage boxes, label it, research it then throw or sell accordingly.
The last point I would make is if there is anybody who could help you accept that help - friends, relatives, neighbours, charities, social workers etc. I wouldn't be surprised if staff at your dad's nursing home have had similar situations with their other resident and may have advice.
Good luck whatever you do - maybe use us on Gransnet if you need a moan, a cry, a scream, any more advice etc. I hope your mum is soon settled and you can get on with your own life.