Gas is reserved
Registered, or it is an offence.
Gas, unvented hot water and water fittings are separate regimes. Holding one says nothing about the other two.
The assumption almost every customer arrives with is that a heating and plumbing business holds a single credential covering whatever they are asked to do. It is an understandable assumption and it is wrong, and the gap between it and reality is where people get hurt. Three separate regimes apply to this trade, each with its own requirement, and competence or registration under one establishes nothing whatever about the others.
Gas is the one people know about. Work on gas fittings and appliances may be carried out only by or on behalf of a business registered with the gas safety register, and doing it without that registration is a criminal offence rather than a paperwork irregularity. What is less understood is that the register, not the card, is the authoritative source, and that checking takes less time than making a cup of tea while somebody waits. Unvented hot water is the regime people have usually never heard of. An unvented cylinder above the prescribed capacity requires a competence quite separate from gas registration, and its installation is notifiable building work. The separation exists for a physical reason: an unvented system is pressurised and stores energy, and where its safety devices are absent, wrongly specified or unmaintained it can fail violently rather than merely leak.
The third is water fittings, and it is the quietest of the three, which is why it is most often ignored. Fittings must comply with their own regulations, certain installations must be notified to the water undertaker in advance of the work rather than reported afterwards, and backflow prevention requirements exist to protect the public water supply rather than only the premises being worked on. A poorly made connection can push contamination back into a shared main, which is why contravention is an offence rather than a private matter between a plumber and a householder. Taken together, the practical advice is the same for any tradesperson including this one: ask which regimes a job touches, ask what covers each, and verify the number on the public register rather than accepting a card, a logo or a confident manner.
Registered, or it is an offence.
A different competence entirely.
On the register, not the card.
Ask. It costs nothing and commits you to nothing.