If someone found an abandoned child at their doorstep nowadays, how would the adoption process work?

I would contact the police. I am British so will expand on my answer based on that. When a person adopts regardless of whether they found an abandoned child on the doorstep or not they have to be approved first. They can't just ask to adopt the child because as it doesn't work like that here and never has.

police or social services would be called, child would enter foster care, case worker would be assigned to find parents and perhaps try to work a plan for them to reunite with the child (depending on the circumstances), if that isn't a good idea the case worker would then contact family members, aunts, uncles, grandparents to see if they would step up to take child in for kinship adoption or guardianship (and termination of parental rights would happen in court or the parents would sign rights over), if noone steps up from extended family or none are found appropriate, then the child would go out to adoption of a homestudy approved adoption family (the foster family may also be asked), the child would eventually be adopted. This whole process could take anything from 6 months to 2 years, all of which the child would be in limbo in foster care and missing out on critical bonding with family. Worse yet, the child may go into more than one foster home during the 'wait' and might experience multiple disruptions and feelings of loss.

I would call police and let them deal with it.