Mrs Lily Wigram, née Barton, would appear to be an interior designer - see
https://www.instagram.com/lilywigram/. It seems that she now lives in Somerset, where her youngest son was born, and I take this to be her facebook page:
https://www.facebook.com/lilybicketbarton/. From this, I have surmised (possibly wrongly, but it seems very likely) that she is the Lily Jane C Barton , mother's maiden name Bicket, whose birth was registered in Westminster in October 1987. Her parents would also appear to have had three sons, whose births were registered in Lambeth, where the births of two of her own sons were registered.
Based on the above, I have concluded that Mrs Wigram's parents are the Nigel A BARTON and Katharine M BICKET whose marriage was registered in Westminster in Q3 1983 (i.e. it appears that her father's proper first name is not Niall). Per company records, Lily Jane Charlotte BARTON was born in August 1987. Her father, Nigel Anthony BARTON, was born on 17 November 1957, and her mother, née Katharine Mary ("Kate") BICKET, was born on 5 February 1958. Her brothers are Ned Alexander Vere BARTON (born 14 XII 1989), Thomas Murray Anthony BARTON (born XI 1991) and Jack William R BARTON (born 1995).
Mrs Kate Barton, née Bicket, is the second daughter of Capt Antony Neilson ("Tony") BICKET (1920 - 19 II 2010) and his wife (married 1953), Rosemary PARR (1928 - 15 I 2001). Her only brother was Simon Antony BICKET (6 XII 1956 - 9 VII 2011) and her sisters are Jane ("Janie") BICKET (born 1954) and Nicola Anne BICKET (born 1960).
Simon Bicket was the husband (married 1987) of Marianne Louise ("M") COLEMAN (born 12 V 1962), by whom he had three sons, Kit Anthony W BICKET (born 1991), John Antony H BICKET (born 1993) and Henry James P ("Hal") BICKET (born 1994).
Mrs Marianne Bicket was the second daughter of (Frederick) Anthony Hinds COLEMAN (21 IV 1924 - 8 VIII 1986) and his wife (married 19 IV 1958, as her first husband), Stephanie MINCHIN (1935 - 4 XII 2009), who was married secondly on 6 April 2002 to (as his third wife) the 3rd Baron Cornwallis. Mrs Bicket herself was married secondly to (as his second wife) the Hon Nicholas Duncan BEATTY (born 1 IV 1961), the younger son of the 2nd Earl Beatty.
Janie Bicket was the first wife (married 1978, divorced 1991) of Christopher John TATE (born 1 V 1953), a great-grandson of Sir William Henry TATE, 2nd Bt (23 I 1842 - 31 XII 1921).
Nicola Bicket is the wife (married 12 VIII 1989) of the 30th Earl of Crawford and (13th Earl of) Balcarres.