Dear Richard,
I fear we’ve got a while to wait for such a conjunction. By my calculations, the next will be in 2164-6 (0.8-degree separation). A graph of the angular separation of the pair around this time is here: https://in-the-sky.org/graphs.php?gtype=8&startday=7&startmonth=1&startyear=2162&duration=8&obj1type=1&obj1txt=Uranus&obj2type=0&obj2txt=Neptune
However, my calculations also suggest there was a recent conjunction of the pair (less close; 1 deg separation) in 1993: https://in-the-sky.org/graphs.php?gtype=8&startday=7&startmonth=1&startyear=1992&duration=7&obj1type=1&obj1txt=Uranus&obj2type=0&obj2txt=Neptune . I’m a few years too young to have observed that, but I’m sure there are plenty of others here who would have done!
Cheers,
Dominic