Reading history books, they often talk about how the Allied Expeditionary Forces were under joint American and British command, but most media (stories and movies and such) generally just show one nation's soldiers at a time, and it seems like the teamwork between nations is mostly something that is happening pretty high up the chain of command (IE, Eisenhower and Montgomery deciding to send a british unit here, and an american division there). In "Band of Brothers" there is a scene where E company rescues some British soldiers trapped behind enemy lines, and another scene where they are escorting a few British tanks, so I am wondering how much stuff like that actually happened. How often were Allied forces coordinating at the 'lieutenant' level versus the 'general' level? How often would enlisted soldiers interact with those from other nations in combat?
Generally speaking (no pun intended) the American and British forces operated independently at the Army or Army Group level. In other words you had essentially no planned interactions between American and British forces at a lower echelon than a three star general.
There are a great number of exceptions however. The first is obviously the border between army groups where British units fought next to American ones. The second is at the staff level where officers coordinated supply needs, who gets to use what roads and so forth. The third exception is the Airborne forces, which often fought as divisions rather than at the Army level, for example, American airborne divisions were under the British Corps and Army control at market garden. Additionally, airborne operations always involve an extraordinary element of chaos and improvisation because of the drop scatter, where small groups of soldiers from various units have to group together and accomplish objectives on an ad hoc basis. So while the scenes in 'Band of Brothers' would have been well out of the norm they are not inherently unhistorical.