The Nimrod is a very special case. In its maritime patrol role it may cruise at a few hundred feet altitude for hours.
In the thinner air at airliner cruising altitudes the max power output is reduced as much as 70 - 75 % and only a lightly loaded four holer can maintain speed and altitude with an engine shut down.
Shutting down an engine will always mean increased fuel consumption because of lower altitude, thicker air and more drag.
The Nimrod also uses much more fuel on two engines at sea level than it uses at 30,000 feet on four engines. Anyway it may spend less fuel with two engines at half power than with four engines at quarter power.
I don't believe in the Danair procedure shutting down two engines on the Comet 4. No way could a Comet 4 maintain a sensible altitude on two engines, and no way could they save fuel on going lower and slower on two engines instead of four engines at a sensible altitude. Even if their straight jet RR
Avons had a slightly smaller thrust loss at altitude compared to modern high bypass ratio turbofans.
It might have been possible for Danair to save a substantial amount of fuel shutting down two engines during descend and keeping the rest at flight idle. In any case they would then have to restart the engines well before the approach anticipating a possible go around in dirty landing configuration. A low compression ratio straight jet like the Avon uses a significant amount of fuel also at idle.
The Comet cousin - the Nimrod - does it at an extra fuel cost only when it is demanded by the mission.