You would sue in the U.K. courts. In common law (The legal system that forms the backbone of both U.S. Law and U.K. Law... and the legal system of enough countries that about ~2 billion people live in a Common Law Jurisdiction) civil suits must be heard in the jurisdiction where they occurred. There are some situations where civil suits can have multiple jurisdictions which allow the plaintiff to do some shopping for a favorable jurisdiction (for example, one can sue for defamation in either the plaintiff's jurisdiction or the home jurisdiction of the speaker/publisher if the defamatory statements were published/heard in both jurisdictions.). However, in this scenario, both individuals were in the U.K. at the time of the incident, and both are subject to likely investigations in the U.K jurisdiction. It's very difficult to physically assault someone when they are on the opposite side of the pond from you.
As to your question about the United States and Federal vs. State Laws and sentencing, it is true that someone who is tried for a crime at the state level can still be prosecuted at the federal level. It's called "separate sovereignty". In practice, it rarely happens, as the Federal Government doesn't always pursue the matter if the states have a trial, regardless of outcome. In fact, when the states have a trial, it's usually a sure sign the feds won't bother (this is part of the official policy on the matter. It's very rare for it to actually happen, and normally when it does, it's because the criminal act involves something that is exclusively a federal matter. Most examples of them doing this are when people who illegally enter the country and are charged with a crime by police.... typically, the feds will charge them for immigration crimes (but rarely for the same crime the state tries them for). It is possible, under exact right conditions for you to be charged for the same crime a maximum of 7 times... but it's not to my knowledge happened.).
With that in mind, a specific state, the Federal Government, and the U.K. are all separate sovereign jurisdiction, so a conviction for a crime in the U.K. does not mean the U.S. won't prosecute you once we have you in our custody. But in order for that to happen, you have to commit a crime in a way that would allow the U.S. to claim jurisdiction, which, remember, the U.S. has an entire holiday dedicated to remembering how awesome it is that they aren't part of the U.K. any more... A crime committed in the U.K. is not typically the problem of the United States... even if it was a citizen of the United States who was the victim.
As for Criminal and Civil matters, in Common Law, Double Jeopardy only applies to specific types of courts. So the results of the Criminal Trial cannot be re-litigated in Criminal Court... but you can sue in Civil Court for monetary damages just as fine (Remember O.J. Simpson. He legally isn't criminally guilty of killing his wife. But he is legally civilly responsible for the wrongful death of his wife on account of he killed her (Best to read that last bit in the voice of Norm McDonald). And Assault is both a criminal and civil tort (Meaning you can sue for damages related to an assault).
Now having established you have to sue in the U.K. keep in mind that the U.S. is one of the few common law jurisdictions that retains the jury in civil court matters. If you sue civilly in the U.K., the Judge will be the person who renders a verdict. Additionally, the process of discovery might be different (Discovery is very broad in the U.S. but I don't know how it compares to in the U.K.).
Edit for additional info:
One thing you will have to consider is that in Criminal trials, the defense refusing to testify cannot be held against the defense during jury deliberation. In Civil Court, the jury (or Judge if no jury is being used) is certainly capable of holding the defense's lack of testimony against them. In both cases, the accusing party almost certainly has to testify, but in the case of the Civil trial where the victim is seeking damages for a crime, they will likely need to testify even if at criminal trial they did not have too. I bring this up because, given the nature of the question, many prosecutors would like to have evidence that is not victim testimony, especially in sexual assault cases, because the prospect of testifying about such a traumatic experience is will can cause victims to not go through with it, forcing the prosecutor to drop the case if the evidence is not week. In Civil matters, since this is about restitution of damages to your person, you need to be ready to testify.
Also bear in mind that the Burden of Proof is different in Criminal and Civil Trials. In both the U.S. and U.K. the burden is a "beyond reasonable doubt" standard, which means that the jury has to have evidence such that there are no plausible ways for it to have happened but the way the prosecutor says it happened. In Civil Court, guilt can be assigned at a lower burden of proof. In the U.S. this burden of proof is called "Preponderance of Evidence" and in the U.K. its called "Balance of Probabilities." In either system, this effectively means that that the trier of fact (fancy term for the person or persons who will determine the verdict, be it a Jury or a Judge) should find judgement in favor of the side that tells the more convincing story. In case your wondering what the difference between the American Standard and the British Standard is, that's largely academic and another example of two nations separated by a common language (Historically, Beyond Reasonable Doubt standard of proof was introduced prior to the Revolutionary War, so the courts in the U.S. still worked like the U.K. Courts. The idea for a lower burden of proof standard for civil trials didn't come about until the 1800s several decades after the U.S. and U.K. split.) The U.S. also has a third burden of proof for court findings between Reasonable Doubt and Preponderance of Evidence, but that's not used in Civil Trials (the "Clear and Convincing Evidence Standard" is used for a number of more boring court matters... it's most interesting use is it's the standard many affirmative defenses need to meet to be successful in criminal court.).
Another final quirk you should be aware of is that depending on where in the U.K. the trial will take place, the finding of judgement might also be a bit quirky. In Scotland, a jury or judge can find one of three verdicts. You have your standard "Guilty" and "Not Guilty", but also add "Not Proven". In practice, it holds the same affect on the defendant as "Not Guilty" but was designed to make it clear that they are only aquitting you because the prosecutor didn't do his/her job properly, namely proving you did it. It's basically letting the court know that the jury thinks that more likely than not, the defendant did what they were accused of, but the prosecutor wasted everyone's time by dragging a case that had plenty of reasonable doubt. Or as one Scottish commenter put it to me (apologies for the faux accent writing): "We dinae (don't) think you did it and dinae you do it again!"
That said, I'm not sure if "not proven" is allowed in Scottish civil courts as it was designed to let the court know that failure to convict was wholly because the evidence provided failed to meet the criminal burden of proof, not because the accused is actually innocent. In Civil court, the could certainly find it's more likely than not he did it.
Suffice to say, that if the defendant was convicted in a criminal trial, then a civil trial is easy as all you have to do is bring a copy of the court case (bring the other stuff too, but really, if it convinced twelve people to stupid to get out of jury duty to convict, that should be evidence enough.). But an acquittal at criminal trial does not mean it won't rise to the level of preventing a favorable civil outcome. Just ask O.J. Simpson. Or google "What does Norm MacDonald think about O.J. Simpson."
