I'll address the Student loans:
Don't beat yourself up too much over the spouse-consolidation. Yup, it was stupid, but nope, you didn't know that at the time. I would be having a come-to-jesus with your wife if y'all have agreed that it's her responsibility to pay, or just suck it up and take over the financial handling. You're also responsible for it, for good or ill, and your financial butt is on the line with them, too.
Because the 120s weren't isolated (90d late in Dec), goodwilling is going to be more of an uphill climb than norrmal (and with SLs it's... a big climb).
Both the 120s and the 90 are major derogatories and their impact doesn't lessen like 30 or 60d lates, so they'll all count. If you were to focus on removal, I'd go for the 90d first. If that gets resolved, THEN start on the 120s. And even though it might or might not help with your HELOC immediately, I'd stay on it to see if you could improve your file. This is a marathon, not a sprint, and some other financial thing might pop up and you could already be in a better position. Future-you will thank you.
You would try to get them goodwill removed - servicers may/will push back, citing the HEA (and they're not wrong), but you could still ask them to remove it.
If your servicer is Nelnet or Navient, I'm not sure it matters (they're tough) - but you might be able to get someone who is having a kinder day to help you out.
As to who to contact? Call up CS for your first volley. Have a script. Stay humble. Maybe cite the impact those derogs are having on your file in your attempt to improve your financial well being in this landscape, bring up the illness, make it personal (something something, heartstring pulling, but don't fib). Even if they aren't as polite as you'd like, you're asking them for a favor, and keep that in mind. If that call is a no-go, find the servicer in question with google, get EVERY single address you can find. Blitz them all with letters.
I got entire tradelines removed. It took begging (but not too much - just honest truth about my journey, being stupid, and needing help). Ultimately I had to wait for some negatives to age off, but still, any progress is worth it.
And seriously, stay on top of those loans - set up an automatic payment just to keep them current if necessary.
Student loans don't report until 90d lates. Throw a monthly account check on your calendar. Even if a payment is initially missed at 30d you have time to catch it before it reports. If there's a financial hardship, talk to the servicer - they'll work with you (to some extent). If these are Federal, you likely have the COVID/CARES forebearance until September/October, but in the future, communicate with and work with the servicers if needed.
Happy practitioner of AZE7or8or9or10 | Team Finances > FICO
