If someone else claimed your child inappropriately, and if they file first, your return will be rejected if e-filed. You would then need to file a return on paper, claiming the child as appropriate. The IRS will process your return and send you your refund, in the normal time. Shortly (up to a year) thereafter, you'll receive a letter from the IRS, stating that your child was claimed on another return. It will tell you that if you made a mistake to file an amended return and if you didn't make a mistake to do nothing. The other party will get the same letter you did. If one of you doesn't file an amended return, unclaiming the child, the next letter, from the IRS, will require you to provide proof. Be sure to reply in a timely manner.
Winner gets the tax benefits; loser gets to pay the IRS back with penalties and interest. The custodial parent almost always wins. The IRS goes by physical custody, not legal custody.
By filing a 2020 tax return, you can claim the 2019-2020 stimulus (500 + 600) for the kids under 17.
As for the 2021 stimulus, if the IRS does not already have a 2019 or 2020 tax return from you, claiming the kids, you won't get the stimulus in 2021 but will when you file your 2021 return, in early 2022.
That begs the question: why haven't you claimed the kids , in the past, so that the IRS has a record of them being your dependent? That is, you may want to file a amended 2019 return to claim them. If you had allowed the non-custodial parent to claim the children, he is not allowed to then let the grandparent claim the children instead (unless the children lived with the grandparent for more than half the year).
