Assuming that the reason your mother baptized you was because she feared for your life, the baptism you have described was both valid and licit (i.e., permitted). When a child below the age of reason is in danger of death, anyone may baptize the child even if one or both parents object:
An infant of Catholic parents, indeed even of non-Catholic parents, may in danger of death be baptized even if the parents are opposed to it (canon 868 §2, Code of Canon Law).
When a child who is given emergency baptism later recovers, the usual procedure is to take the child to a church so that a priest or deacon can “supply the ceremonies” omitted in the emergency baptism and record the fact of the baptism in the parish’s sacramental records. If this was never done for you, that means that your baptism was never recorded. Thus, even though you were validly and licitly baptized, your pastor will probably advise a conditional baptism.
A conditional baptism is not a re-baptism (something that cannot be done), but follows the formula of the original with the exception of this change in the words: “If you were not baptized, I baptize you in the name of the Father, and of the Son, and of the Holy Spirit.” If anything was lacking in the original baptism that prevented you from receiving the sacrament, you will now be baptized. If the original baptism was valid and licit, you will not receive the sacrament again, but the fact of your baptism will now become a matter of record and allow you to receive the other sacraments.
As for your son, all you need do is to request baptism for him.