Was there much overlap between the British Suffrage and Irish nationalist movements? How did women who were involved in the latter cause feel about the right to vote for Westminster elections considering their views on the authority of that body?
This was certainly a hot topic throughout the period for obvious reasons and the only woman elected in the 1918 General Election (yes, just one woman) was Constance Markievicz for Sein Fein (who was in jail in Holloway prison at the time and never took her seat).
Sylvia Pankhurst was expelled from the Suffragettes (the WSPU) for supporting the Dublin Lockout (just prior to WWI) and speaking alongside James Connolly who was executed by the British a few years later for his part in the Easter Rising - a rising that she backed to the hilt and her former comrades in the WSPU denounced.
It seems odd to say that people who were busy planting bombs in letter boxes and smashing windows were also very conscious of being "respectable" but certainly the mainstream of the women's suffrage movement in England actively opposed any link to Irish independence and it was only those who were consciously aligned to the left (like Sylvia Pankhurst) who gave their support, although this support greatly mattered in places like the East End of London that had very high levels of Irish immigration.
My understanding is that Irish suffrage organisations were split on "Home Rule" with some being opposed, some being for and some seeing it as a secondary question, but you're question wasn't about that so thankfully I don't feel the need to expand on a topic I only have a passing knowledge of.
The WSPU certainly saw Ireland as a place they could carry out actions but felt very little need to consult their Irish counterparts.
For example, when Asquith visited Dublin in 1912 four members of the WSPU came over from England and threw a hatchet at him (which apparently grazed him), they then tried to set fire to a theatre he was due to speak at later that day. Irish campaigners for the suffrage were horrified and denied any knowledge of the attacks or the women, who were sentenced to pretty hefty prison sentences as you might imagine.
The frustration of Irish suffragettes with their English compatriots viewing them as either appendages or, worse, non-existent can be summed up by Louie Bennett when she said "We Irish women must resent the independent interference of any English organisation in our affairs. English suffragettes cannot do any good work for us unless they co-operate with us and allow themselves to be guided by our more intimate knowledge of the Irish people and Irish affairs."
Good books on Syvlia Pankhurst include Kate Connolly's "Sylvia Pankhurst" and and "Sylvia Pankurst" by Shirley Harrison - but there are hundreds on the subject obviously.
You might also like to look at Ward's "Unmanageable Revolutionaries: Women and Irish Nationalism" which I read some time ago but thought very good at the time.