The 1st and 2nd battalions were part of the Regular Army. At the outbreak of the war, the 1st battalion was stationed in India and would go onto fight in the Burma Campaign. They were led by Lieutenant Colonel John Harding.

The 2nd Battalion was stationed in Gibraltar as garrison troops. In 1943 they were reorganised into the 28th Infantry Brigade (alongside 2nd King's Regiment, 1st Argyll and Sutherland Highlanders). On Christmas Eve, they were attached to the British 4th infantry and deployed in the Italian Campaign as part of the 8th Army in March 1944. They would go onto fight in big battles like Monte Cassino. The 4th Infantry division were moved to Greece to mediate the Greek Civil War in November 1944, after the German withdrawal.

The 4th Battalion served with the 129th Brigade but never left the UK, instead serving in Kent under XII Corps of Southern Command. The 7th Battalion was formed in August 1939. The 7th was transferred to the 214th Infantry brigade in 1942.

Both the 4th and 7th fought (well) in Northern France after the D-Day landings. They fought valiantly in Operation Epsom at the Battle of the Caen and would later be engaged in trench warfare for a large part of the remainder of the war. They would end up crossing the Rhine.

The 30th Battalion (43rd Infantry Brigade) formed part of the British 1st Army and fought in Northern Africa and Italy

Additionally, the regiment's records are held by the Somerset Heritage Center. 12 battalions were raised:
1st, 2nd, 4th, 5th, 6th, 7th, 8th, 9th, 10th, 11th, 30th and 50th.

the 1st and 2nd were the regular army battalions, 4th, 5th, 6th and 7th were territorial battalions (only the 4th and 7th served overseas).
The 10th was converted to the 7th parachute battalion of the parachute regiment and saw service in Normandy.
As noted by /u/prole_doorstep the 30th were in North Africa and Italy. The other battalions did not serve abroad.