Airways
Musician - I feel the need to tell you that the responses from our Oz cousin do not quite fit the UK scenario. Given that UKAIP reference, you'll find that UK lower airways are usually 10nm wide, but will broaden out into much wider chunks of airspace around the major cities where traffic volumes necessitate more room to manoeuvre.
And, as you surmise, a 10-nm wide airway does allow opposite-direction tfc to be radar vectored when one a/c is climbed or descended through the level of the other. The ATCO will provide radar headings to achieve at least 5nm lateral separation, whilst keeping both a/c within the airway. We are taught to keep a/c away from the edge of the airway too - you never know who is liable to pop up out there !
ATC would not normally take an a/c outside controlled airspace since that airspace provides the protection the pilot has asked (and pays) for. If traffic does leave controlled airspace, then the services usually change and pilots are advised accordingly. More about that in the AIP too.
The upper air routes are deemed to be 10nm wide but the upper airspace is vast and permits greater tactical use. That said, a/c may still need to be kept close to centrelines to avoid areas delegated to military use.