Being a mom and a bike rider are definitely compatible. In fact, I don't think I would have survived motherhood without the stress relief of biking.
I have 4 children. I biked throughout all of their babyhoods, childhoods, and on into their teenage years. It gets easier as they get older, but there's always compromises to be made. When they were babies, I made what time I could to ride. When they were old enough, I took them with me in a trailer. Then onto their own bikes. Unfortunately, none of them have developed a love for riding that I have. So once they were old enough to choose for themselves. they no longer ride. Lazy! They take after their dad, obviously, not me!
That worked out, too. He stayed home with them and I rode. I totally agree with Tinkerbell. You have to decide how you want to approach it. Ride as a family deal, or get out by yourself. You can try it both ways, see what works best for you. Be flexible with your time and it can work well.
Oh, and I did not leap onto my bike and ride home from the hospital! LOL! I can just see that......... my family thinks I'm crazy as it is. But I was back on the bike sooner each child I had. Two weeks after the 4th one was born, I was out there. Overdid it a bit and starting bleeding again, but it was worth it. I backed off to shorter riders after that one, but at least I proved to myself that I was still a rider. And I haven't quite since. That was fifteen years ago. My "baby" is 15. Time goes by so fast! You do need to enjoy them when they are little and when they are growing up. I personally think you can do that better when you have an outlet for your own interests and take some time for yourself, be it biking or whatever you are passionate about.
Okay, I've rambled on enough. This is a subject with so much to say that I need to limit myself.
Time is a companion that goes with us on a journey. It reminds us to cherish each moment, because it will never come again. What we leave behind is not as important as how we have lived." Captain Jean Luc Picard