My opinion on this - it depends. If the demand for DBA's and Developers is low in the area you live in, finding contracts may be tricky. I imagine the best way would be to make sure your name is out there (make a blog, present at SQL events, etc). Once your name is out there, getting the contracts should be a bit easier as you have a portfolio that the employer can look up.
On top of that, setting up a website where people can hire you could help and you could work as a contractor. Having a website with your rates is what companies tend to like.
Alternately, if you know of companies that are tight on DBA's, cold-calling them MAY work, or may just irritate them. I would avoid cold-calling unless you are certain they could use another DBA but lacked the budget and MAYBE they didn't think about hiring a consultant.
But I think that getting your name out there is the big thing. If you come across as "just some random guy" asking for a contracting job and you have no references or credentials to back yourself up, you may not get hired. Or worse, you could get hired for a job you lack the skills for. For example, if all of your DBA work was on SQL Server 2008 with no AG or failover and they want help setting up their SQL 2019 cluster with always on AG's, that may be outside your scope. And from the developer world, that is a very vague term. If you are applying to be a contractor for development work, make sure you list the languages you support. It would suck to get hired in expecting to do ASP.NET development only to find out their language of choice is FORTRAN.
Alternately, there are consulting companies that MAY be looking for DBA's to hire out in the UK. Look for consulting companies to work for so you can get a good grasp of contracting work before branching out on your own. I live out in Canada and know of a few people who do contracting work.
Lastly, if you are desperate for work, find a company you want to work for as a DBA/developer and apply for whatever position they have that you are qualified for. I started out as an assembler (soldering components onto circuit boards) and moved from there to Tier 1 IT (basic troubleshooting of Windows machines and running far too much network cable) to my current DBA/Developer role.
The above is all just my opinion on what you should do.
As with all advice you find on a random internet forum - you shouldn't blindly follow it. Always test on a test server to see if there is negative side effects before making changes to live!