No matter what domain name you have (including .co.uk), yes, you can keep it registered with company A while hosting your site with company B.

All you need to do is to modify the mandatory pairs of name servers within your domain account into your web host’s name servers.

That’s what we called "pointing domain name to a third party web host (web space)" or "changing DNS (domain name server)".

The pair of name servers generally comes in these format:

ns127.hostingserver.com
ns128.hostingserver.com

You should be able to find that name servers info on your hosting panel after you log into your hosting account. Or you can directly check with your web host’s customer support instantly if they provide 24/7 live chat and phone support. If they don’t, email them.

You need not get the IPS tag of your web host unless you want to transfer your .co.uk domain name to the web host. If you transfer, some may incur admin fee in addition to the transfer cost. But since you’re not transferring, simply changing DNS won’t cost you a penny.

For a complete guide on how to point any domain name to a third party web host, you can refer to the source below. It’s very easy to carry out the domain pointing.

Hope that helps.

Wisely
References :
Learn how to point your domain name to a third party web host:
=> http://www.diehardwebhosting.com/web-hosting-tips/point-domain-name-to-web-space.htm

Just for your information transferring of domain name isn’t that expensive. Some hosting companies charge anything from $10-$30 for domain transfer fees.
References :

For .com, .net and .org domains, just log in to the domain registrar’s panel and change the name servers (ask the hosting provider for info on what to put)

For .co.uk you need to change the domains "IPS tag" to the value that the hosting provider gives you. Some US hosts may not know what an IPS tag is if they have never hosted as .co.uk site before
References :

Some hosting company can help you on this. If you are looking for a reliable hosting company, you may check the review here:
References :
http://www.hintpoint.com/green-geeks-hosting-review/

All you need to do is get the name servers from you host and submit to your domain. It is usuailly two addresses:
ns1.newhost.com
ns2.newhost.com
References :

I’ll assume that your new hosting company will run the DNS servers for your newly hosted site. Your current whois listing should list your current provider (or some DNS servers they cooperate with) as your authoritative DNS servers. You will need to change that entry from what it is now to point at the DNS servers for your new hosting company. You should be able to do that update for free. (If your registrar charges anything significant for this, consider another registrar.) Your new hosting service should also allow you to add/edit your domain name and servers (www.whatever.co.uk, mail.whatever.co.uk, etc.) in their DNS system. If they are any good at all, you can do this all from the web.

You will need to coordinate when this is done to minimize the amount of time the world can’t find your servers. That is, you’ll want your newly hosted servers up and ready and the DNS entries at your newly hosted site entered into the hosting service’s DNS. Then make the changes at your registrar so they direct inquires for your server’s IP addresses from the old ones to the new ones. Only then should you shut down your old servers. As it is, any time you change your server’s IP addresses, expect anywhere from a few hours to a few days for all the world’s DNS servers to get the change. What you can do for web servers is to leave the old one up for a while at your old hosting site, but have it redirect all URL’s to the new IP address.
References :

Look through your setting on the registrar site for redirection settings.

Good Luck.
References :