All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
First of all, we advise you to try and persuade one of them in a soft and wise manner to reconsider going out on that day. If this happens, then praise be to Allaah; but if this is not facilitated, then if you can avoid going out at all without causing a disadvantage or harm, then do so.
In case you have no choice other than to go out, then you should avoid making your mother angry, even if this leads to angering your wife, because the right of the mother is a priority as we have already discussed in Fataawa 99471 and 86430.
We would also advise you to settle matters with your wife in secret in the future – without the knowledge of your mother – and vice-versa [deal with your mother's issues without the knowledge of your wife] because it might be that, in general, it is jealousy that leads your wife and mother to act in this manner.
Allaah Knows best.