I am so sorry you have this worry 
Clearly you want the best care and treatment for your mum and both you and she must be confident in that that is what she is getting.
As others have said, be careful about considering private treatment for cancer. NHS treatment is evidence based and follows internationally agreed protocols. Private treatment can be excellent (but sometimes is not) and often not able to deal with unexpected turns for the worse or complications. Also, continuity of care is a much bigger problem in private treatment for complex problems. That is before you even consider costs which can reach 6 figures.
I am not familiar with facilities in Yorkshire or Manchester, so apologise to not be able to be of any practical help.
By all means get advice from a second authority - but also be prepared to be told the same thing.
Non-curative treatment is not the same as 'we've given up on you'. Think of some cancer treatment (without the expectation of cure) as the management of a chronic disease: diabetics are never 'cured' but can and do live healthy lives with their condition. HIV infection used to be a death sentence, we cannot cure it, but people live for many years with the virus.
Do weigh benefits and risks of treatments against each other and against less aggressive options.
You must both be devastated by the news and very frightened - just take the time to recover from the shock of diagnosis and make considered decisions.
Look at the Cancer Bacup UK website, McMillan and Maire Curie websites and gather as much information as you can. Bear in mind that NOBODY, not even the world's best specialist can predict accurately what is going to happen to your mother. All anybody can do is talk about risks and percentages - there is no way of knowing whether you mother might be among the 80% or the 20% of something quoted.
Wishing you the very best of luck.