RAW, this would have been handled differently
What your situation boils down to is whether the Bandits are surprised by the back-shooter or not.
Let's take a look at Surprise on initiative (taken from 5eSRD.com):
A band of adventurers sneaks up on a bandit camp, springing from the trees to attack them. A gelatinous cube glides down a dungeon passage, unnoticed by the adventurers until the cube engulfs one of them. In these situations, one side of the battle gains surprise over the other. The GM determines who might be surprised. If neither side tries to be stealthy, they automatically notice each other. Otherwise, the GM compares the Dexterity (Stealth) checks of anyone hiding with the passive Wisdom (Perception) score of each creature on the opposing side. Any character or monster that doesn’t notice a threat is surprised at the start of the encounter. If you’re surprised, you can’t move or take an action on your first turn of the combat, and you can’t take a reaction until that turn ends.
A member of a group can be surprised even if the other members aren’t.
Now, if all three bandits know the back-shooter is present (and assuming that, like most bandits, everything they don't know is a threat to them) they are not surprised by the attack and combat ensues without any initial strikes occurring.
Why did I say strikes?
Your backshooter's friends most likely know know he plans to flank the bandits. They may not know when exactly the attack may fire, but they're prepared for it and wouldn't be surprised when it finally strikes.
As stated above for Surprise initiative is rolled for everyone participating in this encounter. Each of the bandits that did not know the backshooter was present (stealth vs perception) will be surprised. The first round will begin, as it was rolled. If a bandit that is surprised gets his turn he "can't move or take an action on [his] first turn of the combat". This also means that any member of your party (which shouldn't be surprised) can take their turns and actions normally, even if their initiative puts them before the attack of the backshooter.
The important factor of this situation is who is suprised and not what suprised them. It does not matter if it is a crossbow bolt arcing through the air, the roar of a dragon or a nearby explosion (an exploding keg of whiskey, an illusion or some magic combustion); if you can reasonably rule that something can surprise someone and they are not prepared/preemptively aware of it, they are surprised and can not act or move on their first action.
People often say there are no surprise rounds in DnD 5e; and that's correct, when it comes to the specific term. You don't automatically snag yourself a full set of actions for each ally if you surprise only one enemy creature. That statement, however, significantly complicates how many people interpret surprise.