Domestically there is no real separation between any of the roles in reality. The only thing being a plumber couldn't(shouldn't) fit gas appliances without being Gas Safe registered.
To do any of the jobs mentioned doesn't "require" much in the way of qualification. You can get a Gas Safe Registration as the toy in most popular cereal boxes these days.
It's very common for a plumber to do any wet work, so heating repairs or installations without the boiler, or for Gas safe guys to fit bathrooms etc. Some places your initial training will specialise, ie when I went through my gas board apprenticeship, it was for Service and Repair of Gas Appliances, that's what my NVQ is in. But I think the majority of heating guys dont have NVQ level 3. And its not exactly worth a whole much anyway, plenty have it and still cant do the most basic of tasks.
Even when it comes to renewables, anyone can go out and buy a heat pump and fit it (it wont work well unless its very carefully specified and the system is correct for it) but anyone can do it, so adverts saying boiler/heatpump/repair/install/etc/etc specialist are just in most cases just adverts, theres typically nothing to back up whats "special" about them.
Theres a lot of brilliant engineers out there as well but unfortunately they are in the minority and have no way to distinguish themselves from the next guy in terms of qualifications on titles or the likes.
