I'm no good for very local advice as I live in the other place. However I was recently in similar shoes in a similar place, at 30 years old and buying alone. I managed it only with parental help – a mix of gifts and loans (to be paid back through taking on lodgers) which allowed me to increase my budget as though I earned about double what I actually do, or was going in with a partner earning about the same as me. I'm very happy to talk deets if this is something you would consider, it was a surprisingly painless process.
However I think since you're several years away from wanting to buy and not actually in your expected job yet it might not be that helpful to get too detailed in your plans and projections. A lot can change and your salary, job location, partner situation, local house prices etc etc are not that easy to predict, so the situation you end up seeing at 30 may not be the one you imagine now. You do know the broad outlines of it, which are helpful to reassess against as time goes on.
In terms of actions, I think at this stage you're 100% doing the right thing by living inexpensively and saving as much as you can. Whether those savings become a deposit, a FIRE fund, or anything else, you'll be glad to have them. You might want to consider investing at least some of it in index funds if you don't already, it will hopefully help your money grow faster than your current predictions (which if I'm reading it correctly only factor in your contributions + LISA top up). However exactly how much to invest and in what is a more complicated topic!
If in the future you decide home ownership is your main priority in more of a 1-2 year horizon then you can start to get more detailed, and hopefully things will fall into place one way or the other. Maybe your job/relationship situation will be such that you'll be able to afford the sort of places that currently seem out of reach, or maybe your preferences and priorities will have changed so the cheaper places are more appealing. Or maybe you'll crunch the numbers and say nah, I don't want to move to any of the places I could currently afford to buy, but I am happy with my life as it is and where it is and I'm saving plenty, so I'll keep on keeping on.
From personal experience, when I was seriously looking into buying for my immediate-future self instead of my hypothetical-years-away-self, it became very clear to me what things I would compromise on and what I would not, until I basically had a hierarchy of options that were within the realm of the possible. Basically something like 1) buy a place that meets requirements ABC within budget X, 2) buy at a huge, scary stretch up to budget Y, 3) keep renting a single room, turbo charging savings, and househunting like it's a second job until succeeding at one of the above, which I thought I could cope with for maybe up to 2 years, 4) give up on turbo-savings mode, return to normal-savings mode while renting somewhere that feels more homey, concentrate on career or other pursuits instead. I'm in no way saying you'll find anything similar – just that you'll probably find *something*.
What I wouldn't do is try to second-guess 'well maybe in 5 years I'll wish I HAD (done some incredibly specific thing)'. You just can't know! You're already doing a great job of setting yourself up for success, even if you don't know exactly what it will look like.
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@2Cent: mortgages here are usually capped at 4.5ish times your gross salary. There is also an affordability check for the monthly payments (checking for your normal expenses, any credit card debt, etc) but that's not going to be an issue for OP. Unfortunately lenders also explicitly don't factor in predicted income from lodgers or airbnb as they don't see it as 'guaranteed' income. However, it certainly is real money coming in, and very helpful – just not in securing a higher mortgage directly.