Ring up the CCG and ask when the fast track application was submitted.
The care home may be waiting until they receive payment from the CCG/LA, this could take some time.
After being given the run around by local management about refund of fees paid in advanced I emailed head office saying that from the date fast track CHC was obtained my contract with the home had been replaced by one with the PCT (forerunner of the CCG), I received a cheque by return post.
In your case when you know for certain that fast track was obtained I would stop any future payments and ask for repayment from the date of fast track.
Contacting the CCG could be vital.
You say that somebody in the hospital said your Mum would be fast tracked for CHC. In my case, along with two daughters, I was taken into a side room and told by a consultant that my wife had very little life left, a registrar and nurse were also present. I asked about fast track and the consultant said she would submit the application. I asked about life expectancy and was told that if my wife survived the next 24 hours it would be weeks or months, absolute maximum 6 months, it was 3 months.
After a fortnight with no confirmation of fast track I checked and no application had been made. I asked the GP if he would sign one which he did, submitted by telex in the morning, confirmation by telex in the afternoon.
I then emailed the PCT with details of the three NHS employees present when the statement was made that fast track would be applied for and offering my daughters as witnesses. They apologised for 'confusion in the system' and backdated the fast track.