Saving for a house deposit, how does anyone do this now...

Two ways: minimise your outgoings or maximise your income. Even a combination of both.

However, the catch is that if you do too much of either you’ll be miserable.

Is there anything you can do to increase income? Second job? A hobby that you can turn into a second income?

Have a look at the big outgoings - can these be reduced?
It’s the same now as it was in 70s 80s etc
you have to do without simple luxuries and save your money
Stop in and save up.
and this.

nothing against the op but there's some folk need to give their heads a shake. spending god knows what on 'phone contracts, ticking on a nice new car that they'll never own, memberships of things they'll never use, the latest must have gadgets just so they can flaunt it to others and then twist on because they can't afford a house deposit. it's all driven by advertising that makes people feel inadequate if they aren't living the consumer driven lifestyle.

we stopped in for a year to get our first house back in the '80s.
It is a case of just don’t spend your money on anything else like, the first 5 grand is the hardest, once you see it accumulating though and you can work out roughly how many months you need to press on before you could be in a position to put an offer in it gets easier

Do the lifetime ISA (at least 12 months before you plan to use it)

Have a clear budget, write down all your outgoings and get rid of anything unnecessary, put your money into savings at the start of the month

also stop in and do nothing spontaneously, have a solid hobby you can do in the house for free even if it’s a games console, can easy spend £200 mindlessly on a Saturday on an unplanned trip to the metro centre or something, meal when you’re out, plus popping in B&M for bin bags or something and leaving with a trolley full. There’s no need for this kind of spending that I cut out when I was saving for a deposit

it clicks up quicker than you realise once you get going
I don't even think I was that careful with saving for mine, but I was living with my parents still so it foes make things easier. I had a HTB ISA so put the maximum in that each month plus another couple of hundred each month into a different savings account. Took maybes 3-4 years, had a 15% deposit, with a bit left over, then got the 25% bonus from the HTB ISA. Couldn't imagine trying to save up whilst already living somewhere and paying rent etc.
Also if living with parents, it can give you the opportunity to get a doer upper on the cheap, and continue to live with parents whilst renovation works going on. If you fancied a property development project.