You do need a local valid address in the United Kingdom in order to register for a domain. There are certain domain registrars that offer "Trustee Services"
Here's what I found:
What are the registration requirements for .UK domains?
All second-level .UK domains (e.g. .CO.UK, .ORG.UK etc) registered
prior to June 10, 2014 have priority and the rights to register the
same .UK domain name until June 10, 2019. This is valid as long as the
original domain remains active. Any 2nd level domains that were
registered after June 10, 2014, does not have priority over the same
.UK domain, and therefore the .UK domain name is available to the
general public. If another person holds the rights to the
registration of the .UK domain you want to register, you will not be
able to register that particular .UK domain until after June 10, 2019.
If the requested .UK domain is not registered as another United
Kingdom extension, it is available to the general public. You can
check here to see if anyone holds the rights to the registration of
your new .UK domain. A local address in the United Kingdom is
required. If you do not meet this requirement, you can register the
domain using the registrar's local address at an additional fee (Trustee Service).
- There are no requirements.
Registrant must be a British natural person.
The registrant is required to be a British Internet Service Provider
and be listed as a Local Internet Registry. The registered domain name
must be the same as or a similar variant of the applicant's name.
It's very important that you don't give a fake address because once they find out, they'll take away the domain registration from you. This is because ICANN requires that every information is legit, from contact number to address. If you use a Trustee Service, the registrar uses their local contacts in a specific country to purchase the domain name for you legally using their own business/residential address or whatever else is needed to complete the registration.