I missed this, so to start with original question what ever size the appliance is when fitted with a 13A plug the fuse in the plug limits the power that can be used. I am sure many have used a welding set with small rods which could take well over the 13A but bit at a time allowing the fuse to cool in spite of welder being well over 13A rating the job has been done without any damage or blowing fuse.
So in theory it does not matter what size the appliance is if the fuse does not blow it can be used on a 13A plug.
There is in practice however a problem in that the 13A plug with finger protectors added can no longer get rid of the heat generated by the fuse without damage to the plug.
This is where the use of an oven becomes a problem. Clearly the socket should be installed where the plug will be in free air. Having a socket inside any carcass which reduces cooling of the plug will cause a problem both in the fact it's not getting the cooling and also it's not visible.
When we fit flex the rating of the flex assumes the flex is visible throughout it's length and is in free air to be cooled so with a Baby Belling cooker placed on a counter top the cable is visible throughout it's length and the plug is in free air.
Once we build a cooker into the kitchen then the supply cable is no longer visible throughout it's length. However if an oven is fitted into a carcass and the supply lead is fed through a protected hole as often found in office furniture for computer leads and the like and then into a 13A socket in full view it is unlikely to cause a problem.
As to total loading on a ring final this will depend on what else it feeds. How a washing machine and tumble drier as considered as kitchen appliances I don't know? The Dish washer yes but it holds so little water that in real terms the heater is on for a very short time.
When I first bought this house the tumble drier was in the bedroom until the kitchen extension was built. The extension would be considered as a utility room not a kitchen and the sockets in the utility room come from a different MCB to the kitchen sockets.
For those with a utility room washing machine, tumble drier will likely not be on the kitchen ring final and so we can't assume in the discussion that they will be on the kitchen ring final.
I am lucky when my house was built the immersion heater had both a local and kitchen isolator as as well as the ring I also have two other supplies into the kitchen. Plus a separate supply to utility room. But the kitchen ring also supplies sockets in the rest of the house as does the utility ring.
So since there are so many variations on how houses are wired considering the load from other appliances in a general discussion does not work.
what would people think about such an appliance having a 13A plug and being plugged in to a ring final circuit?
I think if the plug is visible then although not ideal then assessing the risk I think it is acceptable. However I don't feel that it is acceptable where the plug is hidden from view.