I have just had Windows reinstalled on my computer and all the data files were saved.
Did you also save the 'installer' (meaning the compressed archive containing the files required for installation)? What about your license file (if you're using a soft license)?
I have put the Trados files in my new Program Files but cannot work out how to reinstall the program. I have my login and password for my Customer Centre account so it is just a question of reactivating Trados.
Er... no, it's not, actually, as installing a program doesn't just mean moving the program files to a given location on the computer. What appears to be missing are the registry entries created during installation. Note that this isn't a Trados issue: didn't you have to reinstall MS Office, for example?
Do I have to download it again?
Unless you saved the installation files, yes.