In our group wecampaigns in which I participate a familiar is controlled by the player in normal situations; the DM takes over the familiar if the PC orders his familiar to do something very dangerous, gives it a task it may not fully understand (or maybe even misunderstand), leaves the familiar behind, or sends it off on its own.
We always hadlet a familar act "simultaneously" together with its master on a single (the master´s) initative count.
But considering that
- a familiar can act independently
- has - in most cases, because of it´s own dexterity score - a different initative modifier than its master
- can use the "delay action" or the "ready action" which automatically leads to a new position in the inatative order
a familiar should rather have its own initative count.
The problem now is - assuming the familiar is not being carried, sitting on a shoulder, etc - that it cannot move together with its master, if it acts on its own initative count. Even if familiar and master act immediately one after another, someone has to move first.
If master and familiar want to stay together within 5 feet, the only way to accomplish that would be to move no more than 5 feet each round, because otherwise they would lose touch.
Especially concerning the familiar´s Share Spells ability this would have consequences.
Share Spells (PHB): ... If the spell or effect has a duration other than instantaneous, it stops affecting the familiar if it moves farther than 5 feet away and will not affect the familiar again even if it returns to the master before the duration expires. (emphasis mine)