@Legwork
I really don't want to sound harsh here but "GIVE YOURSELF A BLOODY BREAK!!!!".
I think you are overly focused on so many things you think are wrong, sad, embarrassing, failure - they aren't, honestly, they really are not & this can be changed easily in very short space of time.
You did what you did for the right reasons at the time and in fairness,very admirable reasons. To raise your family and care for others. There is huge value in this.
The LinkedIn suggestion is a brilliant one, start connecting with people you knew in the past. You never know what opportunities may come from it.
Write a list of the people you know, they don't have to be friends etc. Could you approach any of them? You have twins, do you know their headteacher or any of their teachers well? Tell them you are trying to return to work and are really struggling with references given a long gap due to caring for family. If they say thry wouldn't know what to write, offer they some examples etc.
Would you talk to some recruiters? Contact them on LinkedIn ask for suggestions.
Are trying are return to work programmes or groups you could join? (I'm not in the UK so don't know them).
Have a very good paragraph to email them.
"HI, I'm X and am considering returning to the workforce after a considerable gap, raising children and caring for ill relatives. I have a Masters on X, several years data analytics experience in Y. I'm very conscious that my experience is dated but would welcome your thoughts on how best to progress, even at any entry level position that I could work up from".
Where I work, retuning to the workforce recruitment is big for all of the following reasons- usually highly motivated, typically very reliable, hangovers and lateness are not ususlly an issue, maturity is very welcome, you are unlike to leave in 12 months time to travel the world, go find yourself or for a minor increase elsewhere.