I found this question in a Stockport Quiz League paper from 2013, when Diane Abbott was named as the first of that time (in alphabetical order); she was also first in November 2021.
This is the definitive source; hopefully the same page will persist and be updated as necessary!
Second on the list in 2021 was Debbie Abrahams, Labour MP for Oldham East and Saddleworth (since 2011).
Third, and the first male MP, was Nigel Adams – Conservative member for Selby and Ainsty (since 2010).
Last on the list in November 2021 was Daniel Zeichner (Labour, Cambridge – since 2015.) Shame really, because he's not one that I'd expect many quizzers to have heard much about. If he wasn't there, last on the list would be Nadhim Zahawi – Conservative member for Stratford–on–Avon since 2010, and Secretary of State for Education since September 2021.
Wikipedia, in its incomplete list of past and present Westminster MPs, has only one entry before Diane Abbott: Charles Abbott, who was a Tory member from 1801 to 1817 (representing four different constituencies, including Oxford University from 1806 to 1817). He served as Chief Secretary for Ireland from 1801 to 1802, and as Speaker of the House of Commons from 1802 to 1817. He was ennobled in 1817 as Baron Colchester.
The aforementioned Daniel Zeichner and Nadhim Zahawi are two of the only three MPs in Wikipedia's (incomplete) list whose surnames begin with Z. The other is Konni Zilliacus, a left–wing Labour MP for Gateshead (1945–50) and for Manchester Gorton from 1955 until his death in 1967. Konni Zilliacus was born in Kobe, Japan; his father was an exiled Finnish nationalist and his mother was born in America. The family settled in England in 1909, when Konni was 15 years old. Following his death he was succeeded as Gorton's MP by Kenneth Marks, who was in turn succeeded sixteen years later by Gerald Kaufman.
© Haydn Thompson 2021