While the Brits were having manpower shortages, this was in reinforcements and contingency operations, such as Greece. They also had limited ability to conduct large grandiose maneuvers like Market Garden, but the could and did put up a good fight.
No doubt they had manpower shortages.
If you've read these posts over the last few years, you can see where Churchill is trying to shift manpower from different industries and services based on the situation in order to get more men into the army.
They were at it longer than us and they were also all over the world, like we were. The difference, we had a larger manpower pool than they had.
In the Bulge, they had a larger combat force closer than the US and able to affect the battle than we had. Monty could have launched an operation as audacious as Patton's, but he didn't have the stones Patton had.
Still, you have to fight the battles with the forces and circumstances as they come. It is easy to Monday morning quarterback what the commanders went through. The difference, we haven't been on the line for 6 plus months working 20 hour days and looking at the casualty list everyday, only to go to a subordinate and tell them to once more go into the breech!
Hat tip to Kipling.