Why doesn't my Lemon Posset thicken? What am I most likely doing wrong?

It sounds like your problem is most likely lack of acid. The acid is what causes possets to thicken. That could happen because the lemons aren't acidic enough (maybe the ones you had the first time were more sour). Re-reading your recipe, I notice that there's a second, simpler potential cause: your recipe simply asks for the juice of 1-2 lemons, and you might just have smaller lemons, or drier ones without as much juice.

For comparison, I've successfully made this lime and lemon posset, which uses 5 tablespoons of lemon and lime juice for 2.25 cups of cream. That'd scale to 42 mL of lemon/lime juice for your recipe. If you think you simply didn't have as much juice as before, or that the volume was less than that, I'd start by simply using more lemon juice.

If that's not the problem, and your lemons are for some reason less acidic, then the main option is still to use more lemon juice. Ideally you'd concentrate it, though, so that you don't end up with a smaller fraction of cream in the recipe. You can juice more lemons, then reduce the juice on the stove down to the original volume. (An easier option would be to add extra citric acid to your lemon juice, if you happen to have it.)

(Much of this was said in the comments, but I went ahead and made it an answer - please don't accept it unless you verify it though!)

I don't think the issue HAS to be acid. Cream comes in a variety of grades, and some contains far more water than others.

If you used a different kind of cream, like one that doesn't have a thicker layer of fat at the top, maybe try boiling a little longer to concentrate the fat and make more of the casein proteins available.

I would say the lemons are too ripe. It is the same with fruit And citris. You make your Jam and Marmalade when fruit is a little under ripe. My last lemon posset wasn't set right through . I noticed my lemons were a little soft. So maybe a little more lemon and reduce it a little before adding to the cream and sugar. My Grandson said Lemon Posset was the best dessert he had tasted better than anything he had had in a restaurant.

I disagree with people saying its not to do with acid, I think the twice it's happened to me I have used too much juice, due to adjusting the recipe from lemon to lime I used more limes, 4 the first time I used limes and 3 the second time as they're smaller and didn't set. It always works nicely when I use 2 lemons.

I have a friend who is a chef so I asked and took her advice and it worked -

To thicken un-set posset: take out the serving dishes and put it in a jug then add some cold double cream stirring as you do so, it should instantly thicken, so you shouldn't need much, it's not an exact science so just add enough until you get the thick consistency you want, pour it back into your serving dishes and chill for a couple of hours and should be smooth, thick and creamy. It has worked for me.

Let the cream cool to about 50,then add your lemons.