I've so been where you are. In fact I kind of an now with dc2.
He only naps for 20-40 mins at a time. Only with white noise and in sling, while moving. Wakes 3-5 times a night. Won't take a bottle so I can't even have a break longer than an hour or two.
It's so so frustrating when they wake up tired and in a bad mood isn't it. I feel like shouting "why won't you bloody sleep?" Several times a day.
It's bloody hard, but with my 1st DC who was the same as yours (good at night sleeping, bad in day) I struggled so much because like you didn't have many friends or family I felt I could call and spend time with. I struggled in mum groups (and I'm a 30 something mum!) but eventually made one friend who introduced me to a few more, it was a lifesaver. I realise if I'm with people, even if he is grumpy all day it makes it more bearable. The isolation is a killer.
So I'd try to focus on that. This is what I'd do if that's any use...
Go round your Nan's for a visit, or your aunt's. Stay all day if you need to. Or get them to visit you.
Go to your favourite café with a sofa and loud music ideally to drown out the whining. Chat to the staff. Stay as long as you can.
My local children's centre has groups especially for younger mums. Have you looked into that?
Ask your HV to introduce you to mums your age in similar situations
Look on Facebook for FB mum groups in your area and post on there to see if there are people who live near you who want to meet up. Or look on mumsnet local?.
I'd also advise to stop trying to get her to nap. Just accept she fights it, and just do what you want or need to do instead of putting your energy into the naps. Accept she will only sleep for 10 minutes and you cannot control it. When she naps tell yourself you've got 10 minutes, have a quick cup of tea and sit down. She sleeps well at night so she is getting some good solid rest, she will be fine.
Get a really comfy sling or one of those hip seats. Ask on Facebook to borrow one if you can't afford to buy. Great for when baby doesn't want to be put down. I literally would not have survived without my sling.
Finally, things really improved for me first time around when I went back to work. It was stressful and only part time but helped so much to have the space and company.
And of course if you are still feeling depressed, do go talk to your GP again.
Good luck 