So as far as I could understand, we first meet Nux in some kind of infirmary with Max hung upside down as his "bloodbag". He can't leave unless he brings his bloodbag along, which leads to Max being strapped to the front of Nux's rig. But then midway through the show, Max detaches himself. And Nux doesn't seem to suffer any problems from losing his bloodbag. Was there an explanation of how that made sense? Or is it sloppy writing? Or did he actually suffer ill effects and I didn't notice?
In this question it's speculated that Nux has some form of cancer, and is therefore retrieving blood transfusions from his "bloodbag" to treat anaemia.
The NHS website states that a blood transfusion can take "between 30 minutes and four hours" to transfer a unit:
Blood is usually given through a plastic tube inserted into a vein in your arm. Each unit can take between 30 minutes and four hours. Depending on how much blood is needed, the whole procedure can take a significant length of time.
We don't know how long Nux had been hooked up to Max for by the time the War Boys were preparing to pursue Furiosa and the War Rig, or how much blood he'd need from Max in order to be restored to a functioning state, or how long it would have been before he'd need another blood transfusion.
My guess would be that he hadn't received enough blood by the time the War Boys were leaving - hence the need to take Max along - but had received enough to keep him going - for however long it was between Max disconnecting and Nux dying - by the time Max was no longer acting as his bloodbag, which is why he seemed to do okay without it for the rest of the movie.