The rules don't treat familiars any differently
Being a familiar doesn't change any interactions with being underwater. The things that matter are whether or not they can breathe underwater and whether or not they have a swim speed.
If they don't have either, as most PCs don't, then you treat the familiar the same as you would any PC when going underwater.
You can read more about the rules regarding underwater movement (PHB, Chapter 8) and underwater combat (PHB, Chapter 9).
Since there is a NPC caster using water breathing on the PCs, that spell could also be applied to the familiars and any other friendly creatures the party would like to join them. If not, then they'll just have to hold their breath via the suffocation rules (PHB, Chapter 8).
Flying speed underwater
As per the rules on swimming, if the creature doesn't have a swim speed, then each foot of movement uses another foot of movement. There is no rule that you can't use your fly speed underwater, just that you use your 'speed'. However, as seen in the Q&A "Can you use the Fly spell to move underwater at a speed of 60 feet?", whether or not you can use your fly speed may be up to you as the DM. The rules aren't clear, so make a ruling as the DM (or talk with the DM, if you are a player) about that.