I need to transfer all the files from an old site into
a new domain, so that the new domain will have the
content of the old site, is there an easy way to do this,
without having to download files and re-upload to the
new domain. Both are hosted by the same host.
If your new site is using the same basic design as the old one, the easiest way I know would be to go to your cPanel and change the name of the folder holding all your site files from Oldsite.co.uk to Newsite.co.uk (or whatever) then change individual subfolders to have your new logo/privacy/about us etc details.
If your changes are more complicated than this perhaps one of the more experienced members can help.
Transferring (downloading and uploading) lots of small files via FTP can take a long while. Compressing them into a big zip file will make the transfer itself happen much faster. You can then unzip the archive at the new host. This can usually be done with the tools in most control panels.
Assuming both hosts use cPanel, you could create a full backup of your account, give your new host access to it, and ask them to restore it on your new account. Not all hosts may agree to do this for free.