That transformer will probably work with a Ring doorbell.
Whether the existing chime does depends on which of the many versions of Ring you get, which version of the software it's using and whether that particular chime is compatible with it.
Some can theoretically control an existing old chime. Others don't have that capability at all and require the nonsense of installing a 'bypass connector' which leaves the existing chime in place but not connected.
The summary of a long and tiresome story is that if you want a Ring doorbell to work as described, you remove all existing wiring, chimes, transformers and anything else. Forget about using it with any existing equipment.
You then buy and install the Ring doorbell, the Ring power supply, and optionally the plug in chime unit. The end.
Very well said.
In my limited experience of trying to fit existing doorbells with ring doorbells, you can match up the correct transformer to serve both, if you are lucky it may work for a bit, but ultimately it doesn't seem to work.
A waste of time.
As far as UK stuff goes, Ring don't really promote their products with existing doorbells or even new doorbells made by someone else.
They don't offer much advice or details on how to do this (UK at least).
More annoyingly, they don't make their own wired chime that you can use as a direct replacement (AFAIK).
As much as I hate the idea, the 'best' way is to obtain a long lead with the transformer built into the plug to power the Ring unit. These don't ever seem to come with the Ring kit. I believe Ring do make them, but often it's just as well to buy one of vast number of Chinese ones available on E Bay.
You could consider keeping your existing doorbell in place, as well as having the Ring doorbell, and have them independent of each other. Far from ideal, and it shouldn't have to be this way, but personally I have found trying to do it your desired way to be a complete drama.
Ring need to get real and think about their UK customers who want to connect their Ring doorbell to their existing one.
(There are some people on YouTube who have claimed to conquer this problem - but the general shoddiness of their work and some of their crazy logic does leave me doubtful.)