My wife came to the U.S. on a J-1 visa as an au pair through Cultural Care Au Pair. She received a weekly stipend from her host family during her program and has never filed a U.S. tax return for that income.

@UserOTP603  Congratulation and wish you a very happy and peaceful married life. Now your questions and answers thereto ---  answers in Italics:

1. Can she still file late returns for prior years using Form 1040-NR?

She needs to file and bring her account current. Because she was NRA -- she is taxed ONLY on US sourced / connected income.  She files on form 1040-NR  ( TurboTax does not support , use something like SprinTax or use a  tax professional to do all the  years --assuming that multiple years are involved )
2. Is there a penalty risk for filing late, and how significant is it typically for au pair income levels?

For late filing there is interest and penalties ( failure to pay and failure to file ). However, while the interest charge cannot be abated, the penalties can often be -- send a letter of apology  explaining that there was no intent to ignore the law etc., requesting abatement.
3. Would TurboTax support the 1040-NR for prior tax years, or do we need a third-party service like Sprintax?

TurboTax does not support 1040-NR, use SprinTax or Tax Professional
4. Given that she is now married to a U.S. citizen, does her filing status or the form she needs to use change?

When did you actually get married ?  AS a NRA married to a US person ( citizen/GreenCard/ Resident for Tax purposes ), she can request to be treated as a Tax resident for the whole  -- you both have to sign  and send in a request ( while filing MFJ ) that you both want her to be treated as a tax resident for the whole year.  Note that  unless her immigration status is adjusted,  this  request is a kind of one way street --- you can cancel / reneg. it only once in her life time.  Of course this will never be your cased.

All my answers are assuming that your spouse  was lax only for one year  ( since Au-Pair s are usually for none months ( I thought ).  If she is in trouble with multiple years, it would be best to seek professional help ( legal and tax ).  IMHO

Does this answer all your questions/doubts etc. ?   Which country is she from ?

Is there more I can do for you ?