What firm do you recommend for buying domain and hosting in UK?
The site is photo gallery in WordPress.
You can register at any register which offers the TLD you're seeking.
You can host at any host which will accept those TLDs (meaning, just about every host around the world).
What you're looking for is a host based in the UK, preferably a good host.
I use WebHostingBuzz for two accounts (US-based) but WHB also has servers in the UK. Since their service has been outstanding (ZERO problems), I can't comment on their support except to say that they must be on top of their servers to keep them error free. Because they also offer "Specialty Hosting," I'm sure that they have WP tailored hosting accounts available (I have a client who required a Joomla account in the US - Joomla is a resource hog so the WHB account tailored for Joomla was a godsend).
:: InnoHosting.com || UK & USA Hosting Solutions || 9 years in business
:: Varnish Accelerated + High end hardware + Low customer to server ratio
:: Small enough to give you personalised support, large enough to cater to your every requirement
For webhosting, I highly recommend Hostgator. I have heard good things about BlueHost too, but have never used them.
Also, you can use their customer service "livechat", before buying, to see which one you prefer.