May I please challenge this?
What are you really upset about?
Are you upset about them forgetting about your birthday or are you upset about their mother and the possibility that they are acting a certain way because of their mother?
And if they are in fact impacted by their mother, then it is their choice to respect their mother’s wishes. Besides, if she is as much of a bully as you say she is, then you should emphasize with the probability that her kids have to end up dealing with her, should they disagree with her. Thus, they might be acting this way to avoid confrontation with her.
Sure, you can cut your online communication with them but that might make things worse because it is an overreaction.
Besides, why is it so important that they wish you a happy birthday? Again, this probably goes back to the issue of what you are really upset about. It seems like your beef is with the mother not with the kids.