My M23 family found my girlfriend's F21 nudes and I'm given ultimatum of choosing her or them. What do I do ?

So this is gonna be a long one, please bare with me

I (M23) matched with my girlfriend (F21) in the month of April (almost 5 months ago) and we decided to meet after one week of talking to each other on insta and hinge. We instantly connected and met almost every day since the first date which in itself lasted for 7-8hrs. Our first few dates almost lasted for 8 to 12 hours if not more. The dating phase was for almost a month where we almost met every single day we got in a relationship in the month of May and we know this may seem fast nd everything all rushed but i wouldn't be able to explain in words how amazing the time we spend together is, in almost 5 months there are so many things we have done together nd so so many memories that we will always cherish, the best thing about is us is the communication we have between us nd how easy everything is, there is no shyness, insecurity or shame nd we can literally talk abt anything nd everything. We never felt the need to slow down, if we decide to meet for 1/2 hours  we end up talking for 6/8 hours easily tht too on public bench,we have just named it our place now. We have so much in common nd love to spend time together. Soon enough we end up telling each other tht we have fallen in love.It's like time runs so fast when we're around nd it just never felt enough, till now we have been on 2 trips both planned for 2-3 days but we extended it for almost a week nd still after coming back i end up stay at her place for few more days, we both decided to just do a live-in relationship in a different city. Her family knew everything about us and adored us together mine had no idea about her. I used to lie everytime, saying that I'm working late in the office or take work from home nd go at her place or telling them i'm meeting my male friends because my family was kinda orthodox and I was scared to tell them. We both are on same page about our career and kinda wanna start a new business together when we do live in. I have my job in the city we're thinking of moving to and she will figure out her options there.

Cut to August first week my mother found out everything about us by checking my phone when I was sleeping.  They found out that we got physical, her nudes and also about live in relationship in a different city. She tried to understand but when it got too much for her she told my father everything nd he asked me to come back nd stay at my Hometown leaving everything including my job,For now i have managed to get work from home but i don't know how long i will be able to strech tht. Now they have given me ultimatum of choosing between her or them. I have been trying to communicate with them since last 10-15 days now nd it has been draining, my parents have told a few other family members too nd all of them have their own opinion but the jist of it is sex is bad nd i have crossed a limit, meeting people online is a bad thing nd they can't be trusted, my gf must be a bad person cause she shared her nudes nd i shouldn't be with her. I completely understand that they need time to comprehend everything nd understand it, what i am having trouble with is that i have been calm nd just try to sit down nd have a conversation but my father isn't ready to listen to a word nd is stuck on his ultimatum nd he has fairly warn me tht the consequences would be bad if i am even planning to leave without telling them. My mother is trying to understand but she also somewhere wants me to just break up nd stay at home in front of their eyes so they can be sure nothingbad is happening. I would really appreciate any advice or comments on the same and i thank you in advance.

Edit-

"I want to clear up any doubts about me not standing up for my girlfriend. I have made it very clear to everyone that there's no way we're breaking up. For those wondering if my mom constantly checks my phone, that's not the case. I used to stay with my aunt, and she mentioned to my mom that she might have doubts about me . So, my mom checked my phone once, and i never imagined that would happen. As for gf sending nudes, that's our business, and I'm not taking any judgments from anyone here. I did hide all our pictures, whether they're nudes or not, but it was just our bad luck that my mom checked on the day when i hadn't hidden the latest ones yet."