When do you decide it is the "right time" to move on to your next role? And when do you think should be the number of lateral moves before a vertical career move? When do you decide to "specialise"?

Love this question! I personally will consider moving once there is no value to what I'm doing generally.

Additionally, you can consider moving vertically when you feel that you have nothing to learn anymore.

The right time is different for each person, but for me I decided to move when it is clear that I will not be able to learn any new things, and there are no progression possible.
I chose lateral move if it is to a different jobscope, or from a less prestigious company to a more prestigious one. If I am at a point where I am accomplished (such as getting good appraisal, headhunted, selected for high-potential programme), then I will choose a vertical move especially if it allows for people-managing, which is a useful skill. I am nowhere near that yet, haha.