Of French portraitists between 1750-1850 the most technically accomplished, and the best creators of 'speaking likenesses' were (in my view) Jacques-Louis David and Ingres. David had a long and productive career in which portraiture was only one of his styles. Of course he is well known for his paintings of Napoleon, but look at his portrait of the lovely Mme. Recamier (and compare it to the equally accomplished likeness of her by Francois Gérard) or his picture of Mme. Sériziat with her toddler son, and you'll see he could show us more than mere likenesses.
Ingres, at least as technically accomplished as David, has something chilly in the perfection of his portraits, in my opinion.
Among British portraitists I think Joshua Reynolds a bit overrated and Gainsborough leaves me cold. The most talented portraitist of your range of years was Thomas Lawrence, and when he admired or was interested in his sitter he was as good as anyone in his era. But he could be slapdash when painting porky businessmen and their social-climbing wives, so his output is uneven.
I'm not a fan of Goya, by the way. Of course he had the misfortune to have to paint, again and again, some of the most unattractive royals ever to occupy a European palace. Even courtly tact couldn't make that sorry bunch look attractive.