I came across this discussion while sorting out a confusion in my list of personal popular song favourites.
To repeat what's been said correctly in this discussion,there are two songs here:-
Anniversary Waltz, music by Dave Franklin (1941/2)
and
Anniversary Song, song by Jolson and Al Dubin (1946), based on the Waltz "Waves on(or Over) the Danube" by Josef (Ivan) Ivanovici (Ivanovice) (1880)

Where people are led somewhat astray in calling the Jolson Anniversary Waltz is that his song is a waltz. That, however, doesn't, as if by magic, change the title. Doing so means two songs get confused

Earlier in the correspondence someone provided the tune of Anniversary Waltz for us to listen to.

Best wishes
Ivor