Hi, Im sorry for your loss. I was in a similar situation to you, consistently achieving good grades and having strong GCSEs, but I had a few personal losses during my a levels and went homeless for a period of time, which lead to very poor results. That is why Im currently doing a gap year and studying 2 new A levels. I had a lot of the same questions you did about taking exams privately, but now I have the answers to share with you.
Your UCAS predicted grades will need to come from a teacher, and you could either ask your previous school to provide them for you, or you could find a tutor who is willing to provide them for you. My predicted grades are from tutors I interacted with on Sherpa, which is an online tutoring site I highly reccomend you check out, though there are many like it. You should probably hurry as you'd want to give your potential tutors as much time as they require to feel confident they can give you a grade before January.
As for the exam centres, you can ask your previous school if you can enroll as a private candidate to resit your exam. If not, there are still plenty of options available for private candidates, depending on your location. JCQ has a list of exam centres here that are taking private candidates:
https://www.jcq.org.uk/private-candidates/ . I wouldnt trust any exam centres not listed on here (unless you're doing CAIE, for which you'd need to use this site
https://www.cambridgeinternational.org/why-choose-us/find-a-cambridge-school/ ). You should also probably check online reviews to find one you feel most secure in
Also, I know its not my place to comment but I would suggest that you should probably tell your parents the truth about your grades situation. They only want whats best for you, and would likely be more than happy to assist you with the costs and with booking the exams. It'll make you feel less alone. As for the encouragement aspect, yes its completely achieveable and im sure you'll do fantastic. People do resits all the time, and if you were achieving A's previously then you already have a more than solid foundation. Best of luck to you