Lots of opinionating about what should happen, and it's all quite good; but the question was really what does happen.
I occasionally see "military news" in the US, and there have been recent cases in which US military persons stationed in Germany have been "involved with the natives." Paternity is a frequent issue.
The picture related here, by those advising US "victims/criminals" is that the situation in Germany is as Wolfgang described it. Once an admission of paternity is made, and a court has accepted it and has ruled "who's gonna pay," facts are irrelevant and there is no appeal. I can confirm that, quite recently, that has been the advice given by the US military to persons in, or intended to be in, Germany.
The situation in the US is much the same, but since each state determines (to some extent) how custody/paternity/support is handled, there is a lot of variation, so it's impossible to state much definitively as "that's how it is."
Once a "paternity" has been ruled on by any court, Federal Law allows the IRS to confiscate any moneys owed to the "responsible party" (tax refunds, etc.) anywhere in the US. Federal regulations also require most employers to honor a "garnishment for child support" issued by any state, regardless of where the earner is employed. This often happens (and can be "encouraged to happen - usually by the mother) by applying for new state aid "because the father fails to pay."
Federal agencies can, and often do support court orders for determination of paternity "across state lines" in cases where a "responsible person" has not been identified. (Rarely, this has been used to identify a mother of an abandoned child.) There is still some contention that a forced submission of DNA samples constitutes "self incrimination," but courts have ignored the argument and it seems unlikely to be successful anywhere. It appears to be rare for a court to order a test if a "responsible party" has already been determined, although it probably could happen in some jurisdictions here.
If any state, local, or Federal agency has made any payments in support of a child for whom a parent "could have paid," whether the parent is identified or is unidentified, any unidentified suspects can be ordered to submit to paternity testing, and moving anywhere else within the US doesn't generally escape the jurisdictional reach of the courts. Once a "parental responsibility" is established by any court, repayment of any amounts paid by "official agencies" usually is ordered. Local jurisdications may vary, but there generally are no "statutes of limitation" on collection of accumulated debts owed to a government agency.
There is probably more consistency between US states now than in the past, since many states were forced to almost completely rewrite their statutes in response to the Federal "Dead-beat Fathers" legislation of a few years back. As is customary, the Fed couldn't order the states to write new laws, but it could, and did, extort fairly uniform compliance by threatening withholding of Medicare/Medicade contributions (the Fed share of the Fed program that states are required to administer and mostly pay for) if they didn't.
Not a legal opinion (IANAL); just my own observations.