Your claim uner EC 261/2004 is against SK as the operating carrier which caused the delay. Neither UA nor AP have anything to do with this and you are wasting your time and pursuing either or both of them.

You are each (however many of you) entitled to delay compensation of EUR 600, thus approx. US$ 660. You were also entitled to a "duty of care" at LHR. If SK did not provide a hotel + meals, claim for those (provide receipts) and local transport if you did not stay at an airport hotel.

No, do not send your case to a bottom feeder collection outfit yet. You have not been denied. Your travel was on 9/5 and you were told 8 weeks which is not the "distant future" by any means. That is roughly 11/4.

If you have not heard by mid-November, call SK and ask. Follow up with this:

Not sure what you sent in, but it may be worth reformulating your claim if you said too much or too little. All that matters is that:

1. SK was the operating carrier which caused a delay at your final ticketed destination, e.g. EWR.
2. The delay was on SK flight XXX on a given date (provide the ticket number, PNR, and passenger name).
3. Provide the flight, date & time of actual arrival at EWR and note the # of hours delayed.

Advise of the claim amount. Omit irrelevant detail (it doesn't matter that you missed work & school).

If you need to pursue, you may need a bottom feeder, simply due to your location.