Care home and housing benefit/ CT benefit

[Goodornot]

Mum is going into a nursing home on Monday. Direct from hospital. NHS funded.

Her house ... rented claims HB and CT benefit.

What do I need to do.

I guess it's not beyond the realms of impossible that she could come back with a care package.

Has your mum had a 'financial assessment' yet. Usually someone from the council will visit and go through her finances and work out if she qualifies for the care home fees paid. At the same time they make sure she is claiming everything she is entitled to.
Most people go into care initially on a six week trial period. She is still entitled to her HB& Ctax reduction for this six weeks while you see if the home is a good fit for her. This trial can go on for 13 weeks if needed. There will be a meeting at some point to decide if she is going to be made a permanent resident- that is when the HB ends and the tenancy needs terminating etc.
It's a lot to deal with but there should be support from the care home/financial assessment people as they do this all the time.

The NHS is funding her care home as she has complex health needs. Continuing care or something

Yes they did say it was for up to 3 months to begin with

HB claimants are entitled to continue to receive HB for upto 52 weeks where temporarily away from home in a hospital or care home (although HB departments often don't know this, or feign ignorance.
So, until the placement is reviewed and a decision made that your mum is remaining in f/time care, she's entitled to HB.
Regarding CT you can apply for an exemption - the local council website should have details.
It's important to keep the council informed of the situation (ie temporary move into care from hospital at the moment and then updating I'd that situation changes.
I hope all goes well and your mum gets the support she needs, and you find dealing with officialdom straightforward.