This is such a challenging situation and I really feel for you. I have been there with my parents, too, many years ago. My dad cheated on my mom, numerous times before they eventually got a divorce. (I am not saying that will happen to you).
I agree with whoever said you should just mention that you saw a few texts when her phone went off the other day (she won't know the content that you saw) and then share how the texts made you feel (worried, sad, upset, hurt?) in a calm and honest way.
However, here's the tricky part. You probably won't have success if you TELL your mom what she should do "Tell dad or I will" or "stop seeing this guy or else I'll tell dad" etc. Maybe you will have more success if you just tell her you saw some texts (but don't share the specific content of the texts), share how they made you feel and then ask her to PLEASE think about this and what she is going to do and ask her to make a plan to deal with this and share that plan with you? I wish you luck. Sending prayers for you and your family during this difficult time.