I know Lieutenants and Captains were landing in the early waves on D-Day, but what about the Majors, Colonels, and Generals that were commanding the battalions, regiments, and divisions. We’re they landing in the later waves, once the beach had actually been secured, or were they in the vanguard of the assault.
I know General Theodore Roosevelt Jr landed with his division early at Utah Beach, but was this an anomaly?
I can give an answer centered on Omaha Beach, since it's probably the easiest one to find extensive sources on. Maj. Gen Clarence R. Huebner, commanding the 1st Division, planned to conduct the initial beachhead assault at Omaha Beach, with Brig. Gen Norman D. Cota, Assistant Divisional Commander of the 29th Division landing with the 116th RCT and assisting Huebner on handling the 29th Division units. Brig. Gen Willard G. Wyman (ADC 1st Div) would also land on Omaha Beach before General Huebner and the rest of the 1st Div command group in order to direct 1st Division elements. So General Officers landing with their troops were pre-planned and hardly anomalies.
Battalion and Regimental HQs certainly participated in the assault. At Omaha, the HQ co of 1st Bn, 116th IR suffered heavy casualties, including Lt. Col Bernard W. McQuade of the 58th Armored FA Bn KIA. Lt. Col John A. Metcalfe survived, as did Maj. Thomas S. Dallas (XO 1st Bn 116th IR), but the command group was pinned down and separated from other Battalion elements. 2nd Bn HQ also came in with the first wave, with Battalion Commander Maj. Sidney V. Bingham among the first to reach the shingle. Lt. Col Max F. Schneider, commanding the 5th Ranger Bn also went in with his men. Lt. Col James E. Rudder leading the 2nd Ranger Bn would also storm the beach, engaging in an isolated action at Pointe du Hoc.
Beginning at 0730, regimental command parties began to arrive, including Col Charles D.W. Canham and General Cota for the 116th RCT. The command group of the 16th RCT landed in two sections, with the first coming in at 0720 losing the executive officer and 35 men on the tidal flat. Col George A. Taylor, who was the Regimental Commander, arrived in the second section at 0815 on the beach.
There would be plenty other Field Grade officers on the beaches, several being killed in action, such as Lt. Col Bernard W. McQuade (58th FA Bn, aforementioned), Lt. Col Thornton L. Mullins (111th FA Bn), and Lt. Col John S. Upham (743th Tank Bn). Other staff officers would also have been present, landing with their respective command groups.
So, while they didn't land at 'H-Hour' with the very first companies, many senior officers landed while the beachhead assault was still taking place, and served key roles in rallying nearby troops under enemy fire. I mean, Brigadier General Cota landing at H+60 is still pretty early!
Don't know a lot about the other beaches, but British Army senior officers seems to have participated in the assault as well. For instance, Lt. Col Nelson-Smith commanding 1st Bn Hampshire Regt, 231st Infantry Bde, 50th Infantry Div participated in the beach assault at Gold Beach.
Source is the excellent 'Omaha Beachhead' (Historical Division, War Department. 20 Sep 1945) accessed via the United States Army Center of Military History. (https://history.army.mil/books/wwii/100-11/100-11.htm) (accessed 2022) The last bit on Gold Beach is from IWM's Nelson Smith, Harold David (Oral history). (https://www.iwm.org.uk/collections/item/object/80012687) (accessed 2022)