Can taking 3 different antibiotics in a month mess up your period or make you late?

Unsure / unlikely. It is unlikely that the antibiotics would have a direct effect on your cycle. The fact that you had a problem requiring three different antibiotics in such a short period of time could have an effect on you cycle, simply due to the associated stress that situation put you under. It is well known that physical and/or mental stress can disrupt a woman's cycle.