"the WSPU (Women's Social and Political Union)" other way round first mention, i.e. the Women's Social and Political Union (WSPU)
"appointed by the Women's Social and Political Union (WSPU) as a travelling..." umm.. just use the unlinked abbreviation here.
Probably worth contextualising who H. H. Asquith was for readers who may not be aware that he was nearly the PM at that time.
"She died of cancer on 21 October 1964" unless somehow notable, we don't normally "do" deaths in the lead.
"at The Women's Library' The isn't part of the title.
"Lancashire on 15' comma after Lancs.
"subsidisdised" I say, what?
Could link Blackburn (since you linked Preston).
"school aged 13, became an" +she.
"hatmaking trade" isn't that more encyclopedically known as millinery? Or hat-making.
in Manchester.[4] In Manchester" repetitive. And link perhaps.
"the Manchester University" well, either no "the" or (preferably), the University of Manchester.
Amended to "studying at the University of Manchester Settlement" with no wiki-link, as the Settlement seems to be distinct from the associated University. (see here and [1] here.) BennyOnTheLoose (talk) 14:29, 7 September 2021 (UTC)[reply]
"a travelling speakers" just one.
"the Labour Party[9] " full stop.
"outside of H. H. Asquith's home" see above. Even just British member of parliament?
Amended to include "Chancellor of the Exchequer" here and in the lead. BennyOnTheLoose (talk) 14:29, 7 September 2021 (UTC)[reply]
Sources I've seen don't say why. McPhee & FitzGerald say it was "shortly after the family moved to London" but no further details; ODNB only mentions it in passing. BennyOnTheLoose (talk) 14:29, 7 September 2021 (UTC)[reply]
" the Second World War " you just had World War I, so be consistent.
"Her articles critical" +were
"then common (attacks on property, for example), " -> "then common, such as attacks on property,"
"held at The Women's Library" the Women's Library.
Many thanks, The Rambling Man. I've amended the article in line with your review comments, with notes on a few points above. Let me know if anything else is required. Regards, BennyOnTheLoose (talk) 14:29, 7 September 2021 (UTC)[reply]