Jack Bartholomew is a nephew of Charles Bartholomew, the Chairman of Wadworth's of Devizes, one of the independent family brewers of Britain. The company was founded by Henry Alfred Wadworth (1852-1929) in 1875, the year in which his sister, Sarah Sophia Wadworth (1853-1938/9 - death registered Q1 1939) married John Smith Bartholomew (1851-1924), Henry's partner in the brewery.
Henry Wadworth had no male issue, and at that time it was unusual for breweries to pass through the female line. So it was that his successor in the brewery was his nephew, John Bartholomew (19 VIII 1882-1 VI 1952), son of John Smith Bartholomew and Sarah Sophia Wadworth. He was married in 1914 to Doris Noeline Cairns (16 XI 1892-26 II 1979), daughter of Thomas Ashley Cairns (1854-3 IX 1908), MP (Liberal) for Newcastle upon Tyne from 1906 to 1908; they had two daughters, followed by a son, Major John Cairns Bartholomew (19 IV 1920-7 II 2016) - Jack Bartholomew's grandfather, who was Chairman of Wadworth's from 1952 to 1996.
Jack Bartholomew's mother, Carolyn, née Pride, was a flatmate of Diana, Princess of Wales, before she was married. Her parents were Barry Wedlock Pride (16 IX 1929-2018) and his first wife, Sally Judith de Pass (born 1934), elder daughter of Lt Col (John) Alan de Pass (1899-1969) and his wife, Rachel Joan, née Brown. Lt Col (John) Alan de Pass was a brother of Lt Frank Alexander de Pass, VC (26 IV 1887-25 XI 1914), the first Jew, and the first Indian Army Officer, ever to be awarded the Victoria Cross; they were both sons of Sir Eliot de Pass (1851-1937).
Rosie Riall's maternal grandfather, Sir William Richard Benyon (17 I 1930-2 V 2014), was the eldest son of Vice Admiral Richard Benyon, CB, CBE (1892-1968) and his wife the former Eve Alice Gascoyne-Cecil (1900-1994), a granddaughter of the 3rd Marquess of Salisbury, Prime Minister 1885-6, 1886-92 and 1895-1902.
As Jack Bartholomew is descended through his paternal grandmother from the 6th Earl of Exeter, this means that he and his fiancée have a common descent from William Cecil, the 1st Baron Burghley. There may be a closer relationship, but I have not been able to find one so far.