Take DNA swabs from one from the babies and use genetic genealogy to identify both parents.

If a person's family has been in the US for at least two generations, we can identify parents and other
close relatives fairly easily. Ancestry.com has a huge American DNA data base.

I think however that the UK doesn't have this kind of data base, the Brits don't do DNA testing
as much. But at least the police could try it and see if they can perhaps get
some family surnames and birth locations of the parents.

of the time.

That number is very odd given the high hit rate with most searches.
Maybe the police genetic genealogists are not well trained.

I myself have done these searches for quite some time for adoptees looking for their bio parents.

Yes Gedmatch is used by law enforcement.
It is quite possible to get enough information from Gedmatch to narrow down suspects from that site.
And there are some workarounds.

Keep in mind the criminal is not the one signing up for site, but relatives who may not even
know there is a criminal in the family.

I think it quite irresponsible of the police to not track down the person who is dumping babies.
Three babies are very lucky to be found alive. Those coming down the pike might not be so lucky.

GDPR forbids the use of genealogy databases by the police (I've just remembered that was a plot point in a British TV drama I watched recently), so they have the police national DNA database, but you're only on that if you have a criminal record. So it needs close relatives with a criminal record. And I think it's saying it therefore only helps 20% of the time.

I haven't read that the police have not tried - was that in the article? I've just read that they have not succeeded in identifying the parents.

I am on all the US DNA sites, including GEDMATCH.
I have been working with DNA and genealogy for 20 years.

Obviously criminals do not tend to sign up for the DNA websites.

When anyone signs up for Gedmatch you are asked if you want to be in the LE database.
Obviously there are non criminals who say yes. I myself opted in because I am not a criminal.
So you have a data base of non criminals. So I am in that Gedmatch LE data base and not a criminal.

Additionally once you see the Gedmatch results for the DNA sample in question,
you can do a bit of workaround and take a look see
at the other big DNA data based on your GEDMATCH information.

LE genetic genealogists are looking for the criminal's DNA matches, these are relatives.
Once you identify the close relatives, by triangulation of the family trees and DNA, you can make a short list of suspects.

For example right now the British police have the DNA samples from three babies.
Very quickly they can determine if the babies are related and how.
They can determine if they are siblings, and full sibs or half sibs.
If half sibs they can determine if the mutual parent is the mother or father.
We can see if the babies are the result of incest.

We could also see the ethnicity of the parents and the location where they probably live
or recently lived.

We can make a short list containing the high probably parents of those babies.

If I was the detective on this case I would at least get off my duff and go to various judges to
see if I could get permission to do a DNA search. The Brits are going to screw around and
let the next baby die when it is is thrown out into the elements. Or perhaps killed at home so
as not to take the chance of detection with the 4th baby.

So when Parliament made the law saying "you can't access genealogy databases to look for criminal relations", that cut off that avenue. And even if a new parliament did decide to pass a different law, it's unlikely it'd be retroactive - ie all the people who did sign up to a genealogy database thinking their data can't be handed over to the police would expect that so continue.