Very sorry to hear your news.
As some who works in palliative care and has seen a lot of people have both private and NHS treatment (plus personally seen my dad go through it) my thoughts to you would be:
There is very little difference in lung cancer in what treatment you will get first line either privately or on the NHS as the situation is usually quite clear cut. Also the best private doctors will all work in the NHS anyway.
Initial treatment decisions are taken by a group of doctors not just one specialist by themselves, which is why it won't vary very much - there will be specialist to interpret the scans, one to interpret the biopsy if there has been one, a surgeon to say whether they can operate and an oncologist to say what they can offer - and usually more than one of each of these too.
The main part of how treatment would very would be seeing your mum in person and seeing how fit she is - most treatments are very tough and so sometimes when you see the person you realise they just won't be strong enough for whatever treatment was being proposed. This is why, even though the Christie are unlikely to suggest somethig different to your local NHS hospital, they can't just say that without meeting your mum.
If you don't have private insurance already, you will find the cost of private treatment extortionate unfortunately. Also although it is usually a nicer environment as WinoHamster says it is terrible for emergencies and palliative care - these both get shoved back to the NHS straight away. We have a local private chemo unit - it is utterly lovely looking but if you turn up with pain, in the NHS a doctor would be found to come and see you, they would talk to palliative care for advice, you would be given pain relief etc. At the private unit, it's literally a chemo factory - they can give chemo and nothing else. My patient was rolling around in agony as they couldn't even give her a paracetamol, just the chemo.
Finally, the best place to have treatment is always the one nearest home. Having cancer, and having chemo both make you feel utterly lousy - a long car journey followed by a crowded waiting room is a nightmare. If your local hospital is small, that's even better - it probably won't be as crowded, the receptionist will remember your names, everyone will be more friendly. You will still get the same quality of treatment - almost all oncologists at small units will also work at the major units like the Christie.
Best wishes to you both
It is a tough time with tough decisions. Your MIL isn't being totally insensitive (OK she might be if she's often like that) as many people do choose not to have chemo, or try it and stop, as when it can't cure you, you don't want to take something that ends up making you have longer but feeling dreadful. It's very difficult to know what to do for the best and each person makes their own personal choice.