No. Assuming this is a standard domestic installation he should should normally be a member of a scheme and can self-certify the installation via the scheme. There is an alternative route, which is to notifiy Building Control prior to installation, then have them inspect the installation once its done but I've never known a qualified plumber do it this way.
Until proved otherwise, work on the assumption that you've been conned and now have a problem on your hands. That means building control, trading standards, solicitor or citizen's advice, etc. Check your buildings insurance to see if you have 'legal cover'. As advised above, try Building Control first. If you're polite and make it clear you're trying to do the right thing I've always found them very helpful.
You want to get it fixed now while you can still use the Small Claims Court to recover costs from your plumber. Don't leave it until you sell the house because it is very likely to become an issue at that point. Also, incorrectly installed unvented HW systems are potentially very dangerous. That's why they are regulated.
