I have never worked it but I have seen a lot of ads requiring recent experience and a current card/clearance.  I believe they are referring to the security clearance card that is required ....I.E.  they do not want to have to spend the bux on the background checks etc. necessary for you to walk through the gate.  Far as certs I have no idea but many folks on here do, I am sure they will fill you in.

You will qualify for whatever you do in the plant. There are so many different processes used and so many different materials being welded with so many different procedures there is no way anyone could have a "blanket" qualification.

There is no special Code required certification for welders in the nuclear field. It is all per ASME Section IX. And yes, you will be tested on the job site.

Security Clearance Basics. What I had to do in 1981 for an outage.

To get in the gate on a nuke you need an "L" (equal to a Dept. of Defense Secret Security Clearance) or "Q" (= DOD TOP Secret). I think I read some where that you have to be a licensed nuke contractor or have some sort of endorsement just to generate the application process. The eQip for DOD security clearance cannot be self initiated. These background checks are very expensive from what I've been told. and so most contractors these days are wanting all "new" employees to have the L or Q clearance in hand. Total Catch 22... gots to have the badge to get hired but cant get hired with out a current (12 months max since last carded into a nuke) badge???
If you are fortunate enough to get a contractor to hire you and go through the security background clearance, do yourself a favor and come prepared. You will need 7 years (10 years for Top Secret) of EVERY employer, place you've lived, with dates and addresses, home office phone #'s, job site/foreman's #, all criminal convictions (depends on the misdemeanor or felony whether you'll be denied). Wife and ex-wives family info, all relatives, professional and personal references, color of sox, brand of margarine last used on your toast...Have I forgotten anything????
Make sure you call all your reference friends and warn them, and don't lie about anything that can be "verified" by the research pukes. You are more likely to be rejected for lying/omitting (trying to bury that embarrassing minor in possession charge when you were 20 y/o) than not. I went through it in 1981 before the internet. I spent a couple of days calling all over the country. Pretty tough for us road whores who can't remember where all we worked last year, let alone 7 or 10 years back. Like I said, this was 30 years ago and I'm sure it has only become stricter in this "terrorist paranoia" we live in now

"but safe operating practices are pretty much standard" Dude, you haven't lived until you get to work with Health Physics Dept (the Nazis on steroids, masochists of all safety depts.). Especially in a Hot zone (radioactive contaminated area such as inside the containment wall).
How hard is the TWIC card?