As I have said a million times, it ain't the same thing at all.
GAS work MUST be done by a Gas-Safe installer, whereas solid fuel is a building regs notification job, but then he isn't installing the heat appliance, just the cylinder. And installing a cylinder is also notifiable work, I think?
Being a competent person relieves you from having to notify and there are other cartel schemes, sorry I mean competent person schemes other than HETAS. Both the APHC and OFTEC do them and there may be more.
So this work is as notifiable as installing a new vented cylinder on any heating system, or at least that is my understanding.
A lot of woodburners have smallish backboilers with 2kW outputs and are only really designed to heat a cylinder. E.g. the Morso squirrel, but even that will boil even a reasonably badly insulated cylinder if there is no heat leak rad or towel rail fitted. Some woodburners have larger backboilers and are really designed for heating. The fact that the owner wants a bigger cylinder would suggest that the present one is overheating, but it may be that merely it does not provide enough baths before it runs cold. This is something the OP should know if he is providing advice.
Anecdotal research suggests the heat leak radiator needs to be 10-20% of the boiler rated output to work. I once fitted a 2kWish radiator to an estimated 10kW backboiler fitted to a Primatic cylinder and it totally prevented the cylinder from boiling even when the stove was alight all day when the owner got trapped in by snow in March 2018.
I wonder how the OP's backboiler is vented. He says that it has 28mm pipes to the cylinder. Is the cylinder Primatic or a direct model? Using a cylinder fitted with a gravity coil as a replacement for a direct one is not going to be a solution unless the boiler is to be fitted with its own cistern, feed, and vent. At this point, I think we are probably in notifiable territory.
A reasonable guess of the backboiler output could be made by comparing the woodburner design with similar models. Precise identification is not always going to be easy and manufacturer information is usually sparse anyway.
I agree that the suitability of the current system needs to assessed. Personally I will not install anything I do not feel to be safe, but I would allow things to be not to current standards provided that the customer is already familiar with the system and that my intervention would be making the system more compliant and not less compliant. I would also advise the customer in what ways the system could be upgraded to meet current standards at the customer's discretion, but would not insist, provided the system is not actually unsafe.
It is worth noting, however, that I have lived in four houses that had solid fuel backboiler heating (none of them fully compliant) and my time spent at the Centre for Alternative Technology means I have a reasonable understanding of what will work and what will not work even though I am not a member of a competent person scheme.
The fact that the OP is asking these basic questions is a matter of concern, but I'd far rather someone asked these questions than didn't ask them.
OP:
1. Can we have a photo/description of the stove or link to visually similar model online?
2. How long has the owner had this system and is it regularly used. If not, why is it not used regularly?
3. Why is a bigger cylinder required?
4. How is the stove vented? Is it connected to a Primatic cylinder, a direct cylinder, or an indirect cylinder, or does it have its own dedicated feed and vent?
5. What is your experience of working on solid fuel installations and on gravity?
6. Is the installation in a hard water area?