Multiple Australian historians (from Ekins, Carlyon and even Bean himself) have cited localised areas in the AIF's area of operations (so "British" sectors) distances as short as 50-60 yards in some cases. Not a Western Front example but the distance between AIF/British trenches and Turkish at Quinn's Post during the Gallipoli campaign were approximately 15 metres apart - whilst not indicative of any kind of "shortest no-mans-land ever" record, it is very indicative that trenches could be and were very proximate in some cases. One of the closest examples I've come across was about 8-12 yards at Zonnebeke.

Ultimately the distance between trenches tended to be dictated by the prevailing terrain in which the trenches were located and thus the "course" they took, remembering of course that a trench system was not just a single winding ditch but a latticework of trenches and "saps" that were both winding in orientation and were dug both roughly parallel and perpendicular to the approximate frontage.