I’m surprised they announced at all. You can at least give them credit for that so you’d know to initiate your go-around.
I got cut off 3 times by 2 aircraft coming into Alton Bay a week ago. First one cut me off on base inside my base as you described. After a go-around, the next one took off right in front of me so I had to do a side step go around. If that wasn’t enough, he made an illegal right turn out almost in to me. Luckily I was watching him the whole time and cut across above and behind when I saw him start a turn into my upwind. And he never even bothered to announce anything. Sheesh.
91.113g
Landing. Aircraft, while on final approach to land or while landing, have the right-of-way over other aircraft in flight or operating on the surface, except that they shall not take advantage of this rule to force an aircraft off the runway surface which has already landed and is attempting to make way for an aircraft on final approach. When two or more aircraft are approaching an airport for the purpose of landing, the aircraft at the lower altitude has the right-of-way, but it shall not take advantage of this rule to cut in front of another which is on final approach to land or to overtake that aircraft.