Can I use a Thomas Cook headset on a British Airways airplane?

There was a time when airline headsets had two prongs. These couldn't be used with your computer, phone or other headset-accepting devices. These days, most airline headsets have a single prong.

If Thomas Cook provided you double-prong headsets, and the BA plane has single-prong jacks, then you won't be able to use them without an adapter. (While you can buy these adapters, they cost more than the headphones BA will probably be willing to sell you very cheaply. The adapters are useful for people who fly a lot with expensive headphones they want to use on a variety of planes.) It could be the other way around, too: you have single-prong headphones you can't use on a double-prong plane. There are adapters for these, too, but again I don't think it's worth your getting an adapter in advance: if your old TC headphones don't work on the BA plane, pay the pound or two for BA headphones and relax knowing you now have all the kit you need for any plane you might fly on in future.

What I expect is most likely is that you have single-prong headphones and that's what the plane takes. If you want to be completely sure about the plane, provide more details of your flight, so that people who care about and know these sorts of things can tell you what to expect on the plane. (To be completely sure about your existing headphones, look at them.)

As for the "may" part of this, if your headphone fits the jack, no airline cares whether you bought it from them on a previous flight, from another airline on a previous flight, or from an electronics store.

You will be allowed to use them, whether they work depends on the plug on the headset and the actual plane you use.
And on whether BA have an in-plane entertainment system. Although it is the norm that long-haul flights have it.

When you know which route you will fly you can ask for one us to tell you whether it is likely that it will be usefull.