It depends what you mean by "Brits", technically during 1916 and the War of Independence we were all Brits! James Connolly was born in Scotland to Irish parents and was one of the leaders of the 1916 Rising, Sir Roger Casement was, as hit title suggests, knighted for diplomatic service to the Crown. There were many members of the "Ascendency", the Anglo-Irish elite, on the Republican side, such as Maud Gonne. Constance Markievicz had technically lost her British citizenship on marrying Count Casimir Markievicz. Erskine Childers was born in England into an Anglo-Irish family and served in the Boar War and the First World War before being elected to Dáil Éireann and becoming one of de Valera's most trusted companions before his execution in the Irish Civil War.

The Irish Volunteers had a unit at Kings Cross in London and these travelled to Ireland before conscription was imposed in 1916. They included many Irish born members like Michael Collins who had been working in London but also many of Irish descent like Joe Good, born in Soho, who would later write a biography called "Enchanted by Dreams". These London Irish stayed on the grounds of Joseph Plunkett's estate and became known as the Kimmage Garrison.

All of these of course have some link to Ireland. There are only a handful I can think of that had no actual ancestral link. Not British but a passing Swede and Finn took part in the 1916 Rising, I posted about this previously on r/ireland. A small group of men defected from the Black and Tans and the British Army during the War of Independence and joined the Irish Republican Army, most notably Rudge Hathaway. This was the alias used by Reginald Walter Stenning. Born in Middlesex he deserted from the East Lancashire Regiment in the summer of 1921, around the time of the Truce, and joined the IRA in county Kerry in 1922. He fought on the Anti-Treaty side of the Civil War and was captured and executed by Pro-Treaty forces on the 25th of April 1923, only a few weeks before the conflict ended. If you search for "Hathaway" on the Military Archives page you will find a copy of an application by his parents for an IRA pension which was refused.