I am looking for a registrar of uk domain names which lets its
customers set up and manage DNS records BEFORE transferring in (to
facilitate the smooth transfer of web + email).
I have found a US registrar that provides this service -
sitelutions.com.  I haven't found a UK one.  Most UK ones will make
your domain name have a period of downtime after you switch the
namservers to them.

If your question is only "domain name",, it's easy to slove.
"domain name" is only relate to DNS Server.

So, you can register an free DNS service first:
(My suggest: http://www.zoneedit.com )

Then, do all of the DNS settings of your new domain name
(A/MX/NS...for web/email/ftp.. etc) at www.zoneedit.com

Then, you can transfer your domain name to the new register.

After transfer is compelete, you can move the DNS Server (NS record)
back to the new register.