Synthetic time series generation according to some distribution
If I understand you correctly, I think I have a simple way and you do not need to overthink it.


Take a Gaussian (Normal) distribution random number generator, it could even be just downloading a set from https://www.random.org/gaussian-distributions/
Multiply by the desired standard deviation, add the desired mean.
From a certain timestamp, change the standard deviation and mean to a new value to simulate the distribution change. This is the same as concatenating smaller sets with different mean and standard deviation.
See if/when your model detects it.