If someone has a cold/flu, does this make their immune system 'busy'. for example, are they more likely to catch a second illness or alternatively are they just as likely to catch it but likely to take longer to fight both illnesses separately?

Depends: This depends on the illness types. Viral illnesses tend to generate a blocking agent called interferon, which generally prevents other viruses from attacking the same tissues. However, interferon does not alter what bacteria might do. If you have the flu, you don't get another flu at the same time, but might get a bacterial pneumonia at the same time.