Public sector organizations
The latest law for UK public sector websites is quite simple. You need to:
Comply with the WCAG 2.1 AA international standard
Publish an accessibility statement
A small number of organizations are exempt, but even these have to comply with the Equality Act 2010 or Disability Discrimination Act 1995 (for Northern Ireland). Complying with the new law will cover you in all cases.
All public sector websites have to be accessible by 22nd September 2020, but there’s a transition before then:
22nd September 2019 – new public sector websites (published after 22nd September 2018) must be compliant
22nd September 2020 – all public sector websites must be compliant
The law for the UK and EU is currently consistent, and it is not expected that Brexit will impact the current legislation in the foreseeable future.
Private sector organizations
In the UK private sector, your legal obligations are far less explicit.
The Equality Act of 2010 makes it illegal to discriminate against people with disabilities but does not refer explicitly to website accessibility. The Act applies to anyone providing a service including public, private and voluntary sectors.