We've given you some suggestions, do you think any of them might help?
I don't know if there is really evidence here of them deliberately trying to exclude you. Because of this I think that instead of sitting on the sidelines and waiting for permission to join them, you could instead invite them both to join you in doing something. You'd soon find out if they were really uninterested in a friendship with you if they turned you down a few times.
They might be really close friends who enjoy each other's company and so do lots of things together, its strange to assume that they ought to be including you in everything they do and mind-reading that you are feeling left out.
That said, if they are deliberately excluding you, then they aren't ever going to be very nice friends for you and it would be better for you to look elsewhere for friends. Unless you live in a tiny hamlet, there probably are other mums who feel lonely and would like to develop the sort of friendships that you want to.
I do think I know how you feel. I don't have close mum friends (IRL, I am part of an active online post natal group!) and do sometimes envy the close friendships that others have developed when they always seem to have someone to chat to at groups. But unless they are a group of Mean Girls deliberately trying to ostracise you, its much more about how you find friends that gel with you rather than there being anything wrong with what they are doing - they are just getting on with their lives and friendships. Its not their job to make you feel included even if that would've been nice.
When I was a bit more depressed, I would have thought and felt the same way you do about this situation. Now that I feel a bit better and more confident, I'd see it as up to me to take the initiative about trying to create closer friendships (either with this friendship group or if you feel they aren't interested then with new people). Put yourself out there, go to a few new groups - better if they are something you are interested in rather than a general playgroup. Be the friendly person who greets new members, suggests going out to a coffee shop/walk to the park afterwards etc. Go on MN local and ask if there are other mums who want to meet up. Contact Homestart (or equivalent) or your HV, and say you feel isolated and ask if they can put you in touch with other mums who want to make friends. if you are religious, then your local place of worship may have a welcoming group that could be supportive.
Try to get away from seeing things like this a personal rejection of you, and more a logistical problem of how to meet like minded people who are also wanting to start new friendships. Its so easy to feel left out, I'm not blaming you for being hurt, I'm sensitive too, I'm just trying to suggest a few practical ways to find friends.
How old are your children? x