I'm 24 and in the same position, OP. I'm training in a career that will have a high wage in a few years but right now, I'm on a low wage as I'm not yet qualified. I don't own a house yet but should have the deposit in 2 years time and I promise it's doable, even if it takes time.
I do understand how difficult it can be to save money, I earn around £1500 after tax and after paying rent to my parents, car insurance, petrol, tax, putting money away each month for things like my car maintenance/MOT, phone bill, essential cosmetics such as deodorant/showergel/toothpaste, cheap/reduced food to make my packed lunches and dinners with etc I can only save about half my monthly income. This is taking into account the fact that I don't go out for dinners, don't have takeaways, don't have holidays, don't have days out that aren't free/incredibly cheap. I don't get my hair, nails or lashes done, I don't subscribe to Spotify or Netflix, I don't have a gym membership etc. It doesn't mean my life is boring by any means, but my friends/boyfriend and I spend most of our time doing things like going for countryside walks, having a small bonfire in the garden during the evening, exploring new places etc which all cost nothing, especially if you bring food from home with you.
Everything I buy is literally essential, I can't find anything I can cut that will allow me to reduce my spending further, my car is my biggest expense but it's a 2009 Corsa so I can't find one much cheaper to own and I can't do without it as I live on a remote farm in the middle of nowhere with no public transport links and a workplace that I'm unable to walk/cycle to.
Like you, I also have a boyfriend and we are saving as much as we can. We have sat down and worked out our expenses that are non-negotiable (bills, direct debits etc), then the other things we need to budget for (petrol, food, MOT/repairs, dentist fees, prescription costs etc) and set aside a monthly amount for each of them. From what was left, we decided an amount that we would put into our savings at the start of each month leaving us with a small amount each month as a "buffer" for unexpected costs. By using this "envelope budget" style of planning costs it means that we always have money for the essentials, we can always afford things like our car repairs etc but we also have a very small amount to spend on things we like while saving enough each month to be able to have a deposit in two years time. You earn more than me, I'm sure there is a way you can reduce the amount you spend and save significantly more than you are now. It's not nice having to reduce the quality of life you currently have in order to be able to save more but it's what I had to do.
As a teen I spent my whole wage each month on clothes, nails, eyelashes, takeaways etc but I sacrificed all of that to be able to save more. It obviously sucks not having the fancy things I did before but I now realise I was living way beyond my means. It means more to me now to be able to see the deposit increasing each month and knowing that I am not wasting my money on things that I don't actually need because I'm building for a future instead. I hope it works out for you OP.