They have no live link to the DVLA so they can't pick up any vehicles owned by the naughty person who hasn't paid a previous fine. However, your name will be on a system somewhere stating that there is an outstanding fine.
In reality, the only way this would get checked is if the local police have reason to run checks on your name. If you're involved in an accident, or stopped at a manned speed check, something like that. Then it might all get very difficult.
Likelihood of it happening - very low. Consequences of it happening - rather serious, probably.