Based on the community-wide bris milah event described in sefer Yehoshua and its analysis in masseches Yevamos, it seems that the Bnei Yirsoel didn’t perform bris milah in the midbar. The explanation given is the danger posed by the procedure, namely the loss of blood (and/or infection).
So, I would conclude that bleeding in the dessert was certainly possible and measures needed to be taken to protect people from its occurrence.