For the three sins of murder, idol worship and immorality, one is to give their life over rather than transgress. If one was in such a situation and transgressed, is it a mitzvah for them to kill themselves afterwards (disregarding local laws etc.)?
You are talking about a category of sins that are so terrible that a person is supposed to allow himself to be killed rather than transgress them (assuming those are the only two options). Once the sin has already been committing, killing himself does no good. It doesn't undo the sin he committed. Killing oneself is also prohibited and he commits yet another sin by doing so.
The Rambam (Hilchos Rotzeach 2:2) writes explicitly that killing oneself is prohibited:
וכן ההורג את עצמו [...] שופך דמים הוא. ועון הריגה בידו וחייב מיתה לשמים
[...] and a person who commits suicide is considered to be a murderer; the sin of bloodshed is upon his hands, and he is liable for death at the hands of God.
There's no recommendation/ obligation/ allowance listed there to kill oneself for committing one of the big 3.