We used Granite Transformations for our worktops. We put down moisture resistant MDF and then had the thin veneer of granite over the top. No one would know and it was about half the cost of solid. It's 6.6mm thick.

https://www.granitetransformations.co.uk/

You might get some alternative options if you look at at Solid Surface. I’m sure Encore by Bushboard do tops that are 38mm thick.

Thanks for all of the suggestions. I'm wondering now whether it might be better to go for the 30mm and just put a sliver of wood between the top and the bottom of the sides of the false dresser top and repaint

I'd suggest you find a local stonemason that supplies quartz worktops, they can fabricate a 30mm quartz so its 38mm thick for you quite easily. If you ever see extra thick granite or quartz worktops in showrooms etc its been packed out with mdf underneath and a front edge has been fitted and the joint line is polished out so its not seen.

I use a local company called Latino MGI http://www.latinogranite.co.uk/ 07807 438950 for all my specialist worktops, could be worth your while giving them a call, i don't know of any companies in Coventry that i could recommend.