The legal situation is that these tests can give you a hint who the suspect might be, and in previous applications of the technique, existing voluntarily-provided genetic information from relatives sufficed to point to the actual suspect. You cannot arrest or convict based on a hint. A hint plus other facts can result in a warrant, which can result in a conviction. This is the case of the Golden State Killer. The details of the case are not generally known, but see this paper for a bit more information. Basically, they used public genetic databases to arrive at a tree of about 1,000 persons related to the suspect, and used other investigative techniques to narrow it down further. Here is the heavily redacted warrant affidavit that will give you some clue what else went into the search warrant leading to his arrest. Simply knowing "the suspect was of Northern European ancestory" is of insufficient specificity to support a search of all full-blooded and partial-blooded Scotsmen.
This "can" happen in the US, but the real question is, how effective is the technique. This is 99% a non-legal question about sufficiently extensive genetic databases. GEDMatch attempted to limit how their database could be used to catch criminals, but had mixed success. There have been attempts to limit police access to genetic databases at the state level, and no laws have been passed. The Dept. of Justice has limited restrictions on the use of forensic genetic data in cases that it is involved with in certain ways, see here. Basically, what they say is
A suspect shall not be arrested based solely on a genetic association
generated by a GG service. If a suspect is identified after a genetic
association has occurred, STR DNA typing must be performed, and the
suspect’s STR DNA profile must be directly compared to the forensic
profile previously uploaded to CODIS. This comparison is necessary to
confirm that the forensic sample could have originated from the
That paper which I mentioned above has detected in SCOTUS some retreat the position that there is no reasonable expectation of privacy in information willingly given to a third party, starting with US v. Jones, 565 U.S. 400. Conceivably, such testing could fall within the scope of 4th Amendment privacy constraints, given a suitable ruling (or argument that results in such ruling). At present, there is no legal impediment to that kind of testing.