Current Harrier operating nations:
UK, India (BAe Harrier)
US, Italy, Spain, Thailand (McDD Harrier)
Other possibilities- Israel have recently expressed interest in acquiring a STOVL fighter (F-35B). The US will sell them practically anything. WI they decided they wanted Harriers earlier? Perhaps during one of Israel's wars, the enemy manage to crater a runway?
Germany. The RAF's Harriers were supposed to operate out of German forest clearings in WW3. Perhaps the Luftwaffe decide they want involvement?
Note that both these countries would be operating Harriers from land bases, which only the UK do currently.
There are many more possibilities for countries with navies operating Harriers. The most obvious is Australia- the British planned to sell them HMS Invincible in the early 80s (pre-82), and it would have come with an air wing of Harriers.
Another possibility is that the countries that operated British-built light carriers (Colossus- and Majestic-class) decide to go with a Harrier-carrier to replace them. IOTL, India did so, Brazil (and France) operate French-built catapult carriers, and the others abandoned carrier aviation. These others are: Canada, Australia, the Netherlands and Argentina.
Canada and Australia are the most likely- the Dutch sold theirs in the 60s, earlier than the others. As for Argentina, they were buying British weapons up to the late 70s (Type 42 destroyers are one example) and in a TL with no Falklands War might have continued. However, IOTL 25 de Mayo (their Colossus, actually the former Dutch one) served until 1997- a bit late.
Some of these can work with no (or more, in the case of Australia) British cancellations. However, in the "ideal" universe, the Harrier is very different- it's the Hawker P.1154 prototype- it's supersonic, a multi-role fighter biased towards air-to-air, and the "Sea Harrier" has 2 seats! The OTL Harrier might still be built- as the Kestrel, or perhaps the Merlin- as a light ground-attack machine, and might be popular for export.
The Navy operate CVA-01 aircraft carriers ITTL, with a mixed air-group of alt-SHARs and Buccaneers, or with all alt-SHARs.