She can start with this. The officiant has 10 days to submit the marriage license to the county clerk. And it may take a few days for it to be added to the database so if this happened fairly recently it may not yet be on the database. Tomorrow is a federal holiday (4th of July) so the office won't be open, although electronic systems should be working.
Also she needs to try variations of her name ie Katherine, Kathy, Katie etc and possible misspellings of her last name if she thinks he didn't know how to spell it:
https://clerk.clarkcountynv.gov/AcclaimWeb/Marriage/FindMyMarriageRecordSearch
To order a copy of the marriage certificate, if one is found:
https://www.clarkcountynv.gov/government/elected_officials/county_clerk/what-is-proof-of-marriage
Skip down to 'how to order'. She'll need some form of US dollars payment.
In my former US Govt career part of my work was 'authenticating' alleged marriages and obtaining proof of these marriages. I've been retired for a few years but this is what I can tell you:
In Clark County an application is required for a marriage license, but IIRC the chapels have that paperwork for the couple to fill out, and it can actually be done electronically these days but that's 'after my time' so I don't know how that works. The actual license is issued by Clark County Clerk's office. ID is required for both parties.
As far as an officiant marrying someone who is obviously drunk or disoriented, unfortunately it's not actually illegal in Nevada for them to perform the ceremony. What IS illegal is issuing an obviously drunk or otherwise 'non compos mentis' person a marriage license. BUT since the license is technically issued by the clerk recorder AND I assume the paperwork is filled out by the couple, the chapel/officiant probably sidesteps any legal responsibility there, too. The kicker is that the marriage is still legal under Nevada law.
A marriage is not legal until the marriage license has been received and certified by the Clark County Recorder's office. If she sees the license he supposedly has, she needs to look for the officiant's signature and (if he alleges it's an actual 'legal copy') a signature and stamp of the county clerk. She needs to look for an actual INK stamp, not a photocopy. They're usually printed on either bond or watermarked paper, but again things may have changed since I retired. The lack doesn't mean they aren't married and it's not unusual for a chapel to give a 'souvenir marriage certificate' for a keepsake or a photocopy of the certificate they're going to file with the clerk. A souvenir certificate is just that, usually a 'fancified' document and a 'pre-recordation' photocopy will not have a county clerk's signature or seal. Chapels also issue 'souvenir certificates' for people renewing vows or other ceremonies where the couple doesn't want it to be 'legal'.
The officiant is supposed to submit the document themselves, but I have known of cases where they just gave the certificate to the couple and said "Here, take this to the courthouse". I had more than one case where the couple didn't understand or forgot and just kept the license. They found out years later when we saw the 'unfiled' certificate that they were never legally married! Those were horrible interviews, I can tell you .I had one case where the husband went home, packed a bag and left her.
Your cousin is going to need to speak to a solicitor RIGHT NOW. In the US you must be a 'resident of the state' to file for divorce or annulment in that state, for example X and Y were married in Nevada but live in Utah. They would have to pursue termination of the marriage in Utah. I assume this means that if it is indeed a valid marriage (ie she finds a valid marriage certificate), she'll need to pursue annulment in the UK. But she shouldn't wait for that, she needs to consult about her home and any rights he has. And if he is not living there, she mustn't let him move in. Personally, I'd change the locks either way. Let him take her to court.
Good luck!!