As I understand it the above (from OllieOnTheRocks) is correct.
The below is based on new goods, I do not commonly purchase used goods in general across international boundaries - but am told the same or similar applies for used goods too now.
So we pay VAT on new goods imported to the UK - this is currently 20% of the declared customs value by the seller on the shipping forms.
International travellers coming to the UK and purchasing new goods in the UK will be unable to reclaim the VAT they pay (as they have done to date) when leaving the country through the abolishment of the VAT reclaim scheme - purchases made before 31/12/20 have a 3 month grace period for reclaims/people to leave UK and still apply. (for another thread, but I understand this accounts for 40-70% of the sales for some retailers of goods at outlet stores at the likes of Bicester Village, so you can imagine if this trade now moves to EU outlets instead due to their schemes remaining in place, then this will be a death nail for retailers of luxury items at boutiques e.g. Omega etc).
For UK travellers purchasing new goods whilst in the EU, you should be able to reclaim your foreign VAT amount from the foreign tax office but that is down to you, the seller and the tax office overseas to resolve, but nevertheless the first point above applies, you will be charged UK VAT when the goods enter our shores.
For UK travellers purchasing new goods in duty free (e.g. Heathrow), well you cant unless you can evidence a foreign residence being your main residence and they are mandated to ship the goods to that address, you can not pick up a watch from duty free without the duty, you in theory can pick up a watch at duty free and walk out with it, if you pay the UK VAT, otherwise in general we do not have the benefit of duty free shopping anymore (except for small items and consumables etc).
In theory also, you should be able to purchase duty free items at EU duty free shops at airports but (and i am not sure about this entirely) and pay no local VAT as long as you allow them to ship it to you to your UK address (much like the new rules introduced by the UK post Brexit deal) then you should also pay no UK VAT as it should be noted as exempt on the customs declaration, however if you take the goods in person then you in theory would have to pay local VAT taxes, and UK import taxes on entry to the UK and it would be upto you to reclaim the foreign element - however the upshot is it is no longer duty free. --- As I say, I am not 100% on this one, maybe someone can confirm, refute? ---
The above was correct at time of writing (or at least what I have been informed by my accountant & tax specialist).
It does however, rely upon you declaring any goods that you are carrying across borders if purchased in foreign shops and duty free shops - that is individual choice and not my place to advise anyone what to do, but it is an offence (tax evasion) if you do not declare items that are required by law to be declared - and I for one, do not mess with 3 people in life...1) My mom, 2) The Taxman, & 3) Lucifer....