In cases where an area was colonised by a democratic nation which inserted its institutions, i.e. the British, and Independence was done without violence, often the ex-colony retains the Parliamentary system along with its civil service and other accoutrements.
Locals have worked for the State, members of the Civil Service, teachers, police officers, etc. They know how the system operates. They are as much a part of the system as were they the Coloniser.
Those who were adults at Independence have lived their lives under the system. Now, taking power, they step in to the posts without dislocation.
They pass the system to their children, and their grandchildren, so the government is basically stable.
In cases where Independence was the result of a war it is likely that the system will be overthrown. And where there was some form of democracy, this morphs into corruption with an elected dictator.
Soon enough all pretence is discarded, and the ‘elected’ leader takes all power and makes all decisions, running the country as his fiefdom.
The route the newly Independent state takes depends on how deep the Coloniser was, how long, and how they left.