Who knows?
BMS can mean anything. Look at the specs to see what it actually contains.
"Common cheap BMS from amazon" likely is comparable to having nothing at all. At some point in its lifetime, the main power switch MOSFETs blow in short circuit, after which it becomes just a piece of wire.
Charging requires active control of current, and a product called "li-ion charger" does that. A CC-CV lab power supply can also do that, if configured right. A common wall-wart won't do it.
BMS rarely includes the charger. Amazon BMS definitely won't. So what they do have is just an emergency security layer in case of haywire charger. Which, on Amazon BMS's is very well known to just blow making a short circuit, not protecting against anything.
But again, this all depends on what the BMS is, is this single cell or multi-cell pack, how the BMS and charger communicate, and so on. This should all be in the product specifications and documentation, but decent BMS and charger systems of course cost orders of magnitude more than random AliZonBay boards.