If you and your son live with your mother, then only one of you can claim the child as a dependent, including all dependent tax benefits including the child tax credit, ability to claim head of household status, and get the stimulus payments. You can't split the tax benefits of the child.
As the parent, you have the absolute first pick on claiming your child as a dependent. Your mother can only claim the child if you allow it and you don't claim the child, and if your mother's taxable income is higher than yours.
For 2020, the only way you can get the child tax credit and rebate is to claim your child as a dependent, in which case you will have to mail your return (since the e-filed return will be rejected). Your mother will have to file an amended return to remove the child as a dependent, and will have to repay any stimulus rebates, child tax credit, or other tax benefits that she got from claiming your son.
For 2021, you can just claim your child on your tax return, you don't have to do anything special even though you did not claim your child before. Your mother can't claim your child if you do, so you and your mother should be having a conversation about your tax situation so it doesn't come as a surprise.
