If I receive a quote for work from a tradesman, and this quote is supplied as a document (PDF) which does not mention VAT as being either includes or excluded, can the tradesman later say that it didn't include VAT and demand I pay 20% more than the amount stated on the quote?
No, a tradesperson is not acting withing the law if they add VAT in an invoice if it wasn't also clear in the quote.
To do so is a misleading omission under The Consumer Protection from Unfair Trading Regulations 2008.
The Pricing Practices Guide section 2, while having no mandatory force, is easy to read and explains to traders that:
All price indications you give to consumers, by whatever means, should include VAT.
There are two things to consider here, whether you will have to pay VAT on top of the price quoted, and whether you will be able to reclaim the VAT yourself. From the tags, I'd guess the latter isn't an issue, but I'll cover it anyway.
If the tradesman is VAT registered (and his/her VAT registration number should be on the quote if he/she is) then he/she can quote with or without VAT, but should specify on the quote. If it is not specified that you should add VAT, it should be safe to assume that the quote is for the price you pay, however watch out for all quotes ex. VAT in the small print of the trademans standard terms and conditions.
The best option would be to clarify with the trademan. A simple "Are you VAT registered, if so could you confirm that the quote is the price inclusive of VAT?" e-mail should suffice.
If the tradesman is VAT not registered, then they cannot add VAT. If they didn't mention their VAT number on their quote (which they should do if they are VAT registered) then this reduces your risk, but it's probably better off to confirm with them.
On the last point, which is unlikely to be relevant to you - If you are not VAT registered then you cannot reclaim VAT even if it was included.