His father, George Joseph Brooke McVeagh, was a landowner and local magistrate serving as the High Sheriff of Meath in 1891.

[8][9] His mother, Grace Alice Maude, was a daughter of a British army lieutenant-general Charles Annesley Benson.

[2] His elder sister Stella was capped for Ireland at hockey and her son, Donald Pratt, became a notable cricketer and all-rounder.

McVeagh scored 3 centuries in succession for DUCC[11] and led the team to the Leinster Senior League title in 1927.

[23] Notable in this period was a series he played against legendary American player Bill Tilden in London where he won 2 out of 3 practice matches against the professional.

[18] McVeagh was at the peak of his tennis career in 1936 when Irish Davis Cup team reached the European zone semi-finals.

[26] On May 14, McVeagh gave Ireland a two-to-one lead over Sweden defeating Curt Östberg in the second round.

[27] On June 5, McVeagh and George Lyttleton Rogers defeated Hector Fisher and William Steiner taking a two-to-one lead over Switzerland in the third round.

[2] However, the Irish duo was defeated in Berlin on June 12 by Gottfried von Cramm and Heiner Henkel.

[30][31] In summer 1936 McVeagh played for Ireland at Wimbledon with B. T. Leader and lost to Kay Lund of Germany and Enrique Maier of Spain.

[32] At mixed doubles he played with Hilda Wallis against the American duo of Don Budge and Sarah Palfrey Cooke.

[33] In May 1937, George Lyttleton-Rogers and McVeagh played at Davis Cup in Montreux and were defeated by a Swiss team.

[44] His practice was very successful and he counted Aga Khan,[2] Sir Oswald Mosley[2] and Edward McGuire as clients.