British forces built to operate while cut off

Did any British military branches face the same risk of being completely cut off as the US Marines?

Yes. Royal Marines, naval landing parties, and airborne forces all planned for missions in which they could be isolated from the main force.

The Royal Marines, Royal Navy landing parties, and airborne troops all expected to operate with a real risk of being cut off. Their missions place them beyond an ordinary land force's secure communications and supply routes, whether landing from the sea or descending behind enemy lines. Operation Market Garden is the conspicuous example of airborne units becoming isolated. Isolation is a recurring design problem for expeditionary and vertical-envelopment forces rather than a uniquely American concern.