Depends what it's conditional upon. If it's something like CRB checks and you're not worried, then relax. If it's dependent on references (which, let's face it, are a touch less certain as you never quite know how the conversation might go), then that is slightly more cause for concern. Really they should have checked references and made an offer BEFORE you handed in your notice. I think it's fair to say that you would like to know that the job is "in the bag" before you hand in your notice. But, that's a fraction too late now. So I would just stay quiet and turn up for work on day 1. IF... if something does go wrong with the references/checks, then deal with it as it arises. But I wouldn't go reminding them that it was a conditional offer, just in case some over-zealous HR bod decides to get the microscope out. So my advice: keep quiet, relax somewhat, and plan on turning up for work on day 1.