Sorry to hear about your father
@blueorchid It's obviously a worrying time for you, but in terms of finances, as your mum has been placed in emergency respite by social services the best thing to do - if they haven't already done so - is to request a care needs assessment and explain the circumstances with regards to a lack of POA. Did your parents have joint bank/savings accounts? With regards to needing ‘authority’ to arrange the funeral, as far as I’m aware there is no actual authority required. If you mean are you able to withdraw funds from the bank in order to pay for the funeral that may be a problem but suggest that you discuss this with the bank and explain the circumstances. You are right that in the absence of POA someone will need to act as Deputy for your mum and I understand that this could take several months to arrange.
With regards to paying your mum’s care home fees, the state does not seize all of your mum's assets to pay for the home straight away. Does she have any other assets other than the house? In usual circumstances, if she has savings in excess of £23,250 she will be deemed a self-funder and in a position to pay for her own care fees. The situation is different as your mum will not be able to access her funds so speak to social services about the situation as they will be able to refer to the financial assessment team for advice.
Following a care needs assessment the local authority conduct a financial assessment and if your mum has savings under £23,250 the local authority will contribute towards her care fees but her house will be disregarded as an asset for the first 12 weeks, to enable it to be sold. The other option is to enter into a deferred payment agreement with the local authority if the house is not sold within 12 weeks. This involves them placing a charge on the property and paying the care fees until the property is sold or your mum passes away - this will involve an arrangement fee plus interest added to the repayment amount. There’s a lot to think about but in the first instance speak to social services to explain the situation. It may also be helpful to speak to the Dementia Connect helpline to talk things through:
https://www.alzheimers.org.uk/dementiaconnect
This factsheet about paying for care may also be helpful to you:
https://www.alzheimers.org.uk/get-support/legal-financial/care-home-fees#content-start