You do any research using something like TOR. Use it on one device only and take it with you or completely destroy it
Oh you can order a birth certificate easily enough.
I don't know about checks now but it used to be your ordered a BC of a child born about the same time as you that died within the first year of life so there was unlikely to be a paper trail.
Once you have that you can get a passport. There might be more checks now, particularly if you are ordering your passport for the first time as a 40 year old.
You would have to forge the signature of the person saying the photo is a true likeness. I suppose you could use your real passport to sign.
Initially head for somewhere there are lost of people moving in or out of the area and where cash can be used.
Blackpool springs to mind, lots of B and Bs that even today can be had for £20 - £25 a night and will accept cash.
Or find a job with accommodation.
Have some excuse for no recent references (possible prison, that's a risk that they don't want you but cold be worth the risk, looking after an elderly relative who is now deceased or you have been living and working abroad).
There are a few things you cannot hide, eye colour (although you can get coloured contacts) medical history, depending on whether you have any chronic conditions.
You will need a NI number, this could be tricky, easier if you are 18 and have never worked, you should still have one but you could have lost the card. The 'I've been living and working abroad' could work. Again that would be easier if you are younger, maybe you emigrated with parents in your early teens and have just come back.