I wrote about the care work people are
already doing in the current system without UBI.
Not some altruism they would start doing in a hypothetical system.
It's already being done in large quantities, often in addition to having to work to survive. My argument is that since people already do a large amount of unpaid work (in addition to a paid job often), they won't stop doing it on UBI.
That you feel the need to put care work in scare quotes says to me that you probably live in something of a filter bubble about this, because when I think about this issue with regard to the people I know or have met over the years, it seems to be obviously a very large number of people.
However, since you asked for statistics, here are some reports:
> https://www.nice.org.uk/news/article/unpaid-carers-need-more...
> Approximately 6.5 million carers in the UK are unpaid with 3 million balancing work with caring responsibilities.
> It’s estimated that unpaid care saves the UK £132 billion a year in care costs but a recent report by Carers UK found that more than two thirds of carers are using their own income and savings to cover the cost of care and two in five say they struggle to make ends meet.
That's 6.5M out of approximately 40M working age population. I can say from experience and familiarity with the field that it's not trivial amounts of time and labour either. If it were paid, it would be a significant second job.
Those figures are pre-Covid. Since the start of the pandemic they have increased:
> https://www.carersweek.org/images/CW%202020%20Research%20Rep...
> There are an estimated 13.6 million unpaid carers in the UK today. Most of these unpaid carers, 9.1 million, were already caring before the coronavirus outbreak. A staggering 4.5 million people have started providing unpaid care since the outbreak. This represents nearly a 50% increase in the number of unpaid carers since the crisis began.
> An estimated 26% of the UK adult population is providing unpaid care to an older, disabled or ill relative or friend – that is equivalent to one in four adults.
The above figures refer only to caring for the ill, disabled and elderly (generally called "carers").
It does not count caring for children which is, of course, also unpaid but valuable work that presumably would continue in a UBI system and is also done by a large number of people.
Based on the above I estimate (very crudely) that the number of people engaged in significant time performing unpaid but valuable labour in the form of care for children, ill, disabled or elderly people is on the order of 30-50% of the adult working-age population in the UK.
I would expect them all to continue doing so under UBI... And the ones who are currently conflicted by insufficient time to do both a paid job and sufficient care work for the perceived needs (there are many people in this tight spot) would, of course, increase the latter form of labour under UBI.