The accessibility regulations are contained in the 'Public Sector Bodies (Websites and Mobile Applications) (No. 2) Accessibility Regulations 2018'.
Basically you have to meet the Web Content Accessibility Guidelines (known as WCAG 2.1 level AA standard).
This applies to mobile phones as well as normal websites.
A bit of care is required in choosing suppliers. One that we have been looking at promises all the above, but when putting some of their example council websites through some testing routines, they failed the checks, but the provider has cunningly added a widget that more or less then meets the criteria. It's not perfect, there may be some screen readers for example that won't work, but it will probably be acceptable. It may also fail to work on older hardware / operating systems.
So we will be looking for other quotes.
However accessibility is not just about technical issues. Colour schemes are also very important so that there is a good contrast between the text and background, and the examples we looked at above failed on those grounds alone, probably because the council specified some corporate colour scheme.
Some of the best examples are government websites of course.