Kids at reception?!

As long as your nieces and nephews are in your wedding party in some way, it's okay to include them and not other children.

Just address the invites to "Mr. and Mrs. Smith" instead of "The Smith Family," and spread the word via word of mouth that the only kids invited to the wedding are those in the bridal party.

It's completely fine to invite your siblings children and not extended family or friend's. You're correct that just not mentioning kids on the invitation would be the proper way to do it. Keep in mind that when addressing invitations, if someone has a significant other, you should call them by their name, not "and guest."

I'm in the same boat! We were intentional in addressing our save the dates to the specific family members it was intended for, and then we also included an FAQ section on our wedding website. The first question we have on there is, "are kids welcome?" and our answer was, "We respectfully request that the ceremony and reception be adults-only." I feel like that's as nicely as we could put it, but this has definitely been a tough dilemma for us as well!

I was in the same position, even the kids on my family and kids from close friends arent invited, so what we did is that we put on the invites the spaces reserved, so it would be clear that they can't show up with extra people or little people... For some people can be rude, but in my cae 2 things happen, first the people invited to the wedding know we don't like kids, and second... weddings aren't for kids.

As an advice, it could be difficult, but you should talk to your mom about the gest list, my mom tried to invite her friends, but you should invite the people that really meant something to you and your spouse, sometimes it's not about the cost or budget, it's about limits and the people you want in your special day.