Hi. I've done a few flights this year with oxygen, and a few more to go - here's what I can tell you:
BA will allow you to use the on-board oxygen bottle (not the drop-down mask) if you book it in advance and get a doctor to fill out a form. However they will stop offering this from May 2018 which will bring them into line with most other airlines.
All airlines allow P.O.Cs on board. They will have a list of approved brands. The most common are Inogen and Airsept. Rent one from healthoxygen.com/rental-inf... for around £250 per week. They are very knowledgeable there. P.O.C.s are much easier than oxygen bottles. You can get P.O.C.s on the NHS but you're not supposed to travel with them.
Each airline has a different policy on what notice you need to give them before you fly if you are bringing oxygen, whether bottles or a P.O.C. (No one will allow Liquid Oxygen).
For example EasyJet allow P.O.Cs on board and you don't have to give any notice, however you do need a doctor's letter certifying you have a genuine medical need and that you are stable.
Google your airline name plus the word "oxygen" and you will find the part of their website which tells you their policy.
Sorry if this is all old news to many of you, but hopefully it sums up much of the above.