SMS is interoperable between every phone and phone company on the planet.
When I (UK) switched in the late 2000s, it took about an hour.
I've read some porting horror stories online over the years. Yours is probably something like the 6th or 7th. I assume things do sometimes break, but it doesn't seem to be particularly frequent, considering millions of people switch carriers every year.
I will say this though, switching mobile goes rather well almost all of the time. Porting numbers on fixed telephone lines is much more apt to fail.