If there is any benefit to having a .co.uk domain the UK, I believe it is negligible.
I think its best to use a domain that is considered country neutral by Google, see:
https://support.google.com/webmasters/answer/62399?hl=en
Under:
More about domain determination
If you use a generic domain, if you ever want to go international (site for cifferent countries and/or languages), that domain will work well, but if you use a .uk domain you'll probably need to get a new domain; and that is an issue. I started a site that went international: from UK & Ireland to: UK, IRE, US, CA, AU & NZ. Then late last year the site also went multi-lingual: EN, ES, FR, DE, IT, PT & SH. I started with a .co.uk domain, but I had to move to a generic domain (like .com) once the site wanted to rank in different countries. The result is I have two domains with about links from about 3,500 domains split between them. And I have refrained from 301 redirecting them, as this may affect traffic from the UK, which is 60-70% of income.
However, if you think its highly unlikely that you will be going international, I would say it matters little.