Many of the brushes you see people use online to spread glue will actually be disposable. The type I'm thinking of are something like 50c each, and although they can be cleaned and reused they're cheap enough that probably the majority of users are comfortable using each brush once and discarding it.
but when the gluing setup is finished and one has time to clean the brush it is already set.
Most PVA-type glues have an 'open' time of between 20 minutes and about an hour, depending on local conditions. Even if you're doing a complex glue-up and you can't leave the work to get the brush to a sink for cleaning you could have a jar of water near the workbench to dunk brushes into to stop them from drying out until you've finished clamping up.
Once you get to rinsing out the brush it's advisable to use warm water, it really makes quite a bit of difference compared to using cold water; so where available use warm water. After thorough rinsing, washing the brush with soap and water can help remove the last traces of glue although this isn't absolutely needed.
While you can use brushes of nearly any kind to apply glue I would recommend you confine yourself to those with synthetic bristles. Synthetic brushes are generally stiffer and easier to control (nothing is worse than trying to spread glue accurately on a joint face using a very soft and bendy brush, I speak from experience!) and their bristles are more forgiving of trace amounts of glue being left in them from incomplete rinsing.
In addition to paintbrushes silicone rubber brushes (intended for basting meats or glazing pastry) are used by some woodworkers to apply glue. Although these are great in that hardened glue doesn't stick to them and it can just be peeled off the next day, I don't like them because they're too floppy. But silicone rubber spatulas on the other hand are well worth experimenting with as glue spreaders, they work brilliantly IMO.