The bride pretty much always organises the wedding and it's from a bride-centric point of view.
From your post it seems like the bride does not have a close or special relationship with your children. So why would your children be included?
The communication was also not clear. You mother offered to pay for the bridesmaids' dresses without clarifying they would be for your children?
Why did you and mum all assume your children would be flower girls, if you're not paying for the wedding and you don't have a close relationship with the bride?
Certainly not close enough to be discussing the wedding directly with her and directly asking either bride or groom whether your children would be involved in the wedding.
And willing to bet the bride is more emotionally close to her cousin's children than yours.
And I bet that's why the bride wrote such a lovely note - so you and mum would not be offended.
I'd suck up the disappointment, find a good babysitter and go have a romantic time at the wedding with your husband.
Now that the mistake is realised and out in the open - new bride and groom will probably love and respect you a whole lot for not making your diappointment their pain in wedding planning.