Lawyer here, used to practice family law.
First, the birth parents would not even have the option of trying to get their bioson back after the adoption went through legally. All their parental rights would have been severed. They might try to seek out a relationship with the child, but the adoptive parents would have the option to refuse. What you probably have in mind are the cases like Baby Jessica, where a child went back to bio parents after an adoption. In those cases, the adoptions were not done properly.
If the adoptive parents named the bio parents in their will as the guardian of their child, the bio parents would get the kid, and they could try to adopt him so that they would have full parental rights again. The adoptive parents’ relatives might try to fight this, trying to get custody or even adopt the child themselves. Those situations have a decidedly mixed result in the real world. The standard is, “what is in the best interests of the child?” Different judges will have different opinions, and all cases are different. The outcome would depend greatly on the relative circumstances of the bio parents and the adoptive parents’ relatives.
Factors that might be considered:
How old are the relatives seeking to adopt? If they are elderly, this may be a concern.
The health of the parties concerned.
The home life the parties could provide.
a. If someone could provide a stay-at-home parent, that would look good to many judges.
b. Money comes into it, though usually only to the extent of being able to provide a comfortable life–wealthy v. middle class is usually not going to be a determining factor.
Previous child abuse allegations, criminal record, etc.
How willing the parties are to maintain good family relations with the family the child already knows. If the bio parents would maintain contacts with the child’s adoptive relatives, big score in their favor. If the adoptive parents’ relatives would cut the bio parents out (if they had a relationship with the child), big negative against them.
If the bio parents would raise the child in the same religion the adoptive parents did, usually this is a good point for them in an adoption.
Child’s preference (if old enough to say; usually this starts around age 7).
Who did the adoptive parents pick as guardian? That they picked the bio parents is a big point in their favor.
Those would be the biggies.
Now, if the adoption is uncontested, there would almost certainly be no issue that the bio parents could adopt their bio kid if they were not unfit. The adoptive parents picked them as guardians. That’s usually good enough. Also, if the bio parents adopted before, they’re almost certainly fit to adopt again.
Also, when parents write their wills, they often specify that their money should be placed in trust for their child, so their guardian can use it to raise their child. Frequently, the guardians are named the trustees, which gives them some significant power. Even if the adoptive parents’ relatives got the child, it would not affect the trust arrangement. Bio parents would have involvement at least in the financial portion of affairs.