Doomed Timelines are the result of failed events that needed to take place to perpetuate the existence of Paradox Space. Not everything that happens in a timeline has a chance to doom it, only the ones that are part of a fixed time loop (such as when time players utilize their alternate selves teleporting through time, when the players are created through ectobiology, or the creation of first guardians). If these events do not take place (such as the time player not traveling in the correct way to close the loop, the ectobiologist not creating the players, or the first guardian not being created) then the timeline is cut off from the alpha timeline and 'doomed.'
Other than one of these events taking place (or rather, not taking place), it is unclear as to how significant the change has to be in order for the timeline to be doomed.
It is possible to communicate between the doomed timelines by the Seer class (Terezi accessing her Game Over self's memories to determine the preventing of Vriska being allowed to challenge Bec and Rose using her dream self to inform Davesprite Dave about the state of their timeline [though Dave explains at one point that dream selves exist somewhat out of the normal time continuum, allowing for memories to flow between doomed and alpha timelines]) and also through the use of time travel (Davesprite Dave traveling back along his timeline to a point before the timeline was doomed in order to prevent the timeline ever being doomed in the first place), by Squared Sprites (who gain access to many, if not all of their alternate selves memories), and through the visiting of dream bubbles and seeing the members of doomed timelines.
With the case of Lord English he is not summoned by the creation of a doomed timeline but by the ending of a universe, which occurs upon
the death of Snowman by the hands of Noir, and thus ending the trolls universe.
tl;dr: There isn't a set point exactly for determining when a timeline is doomed, but we do know that is a closed time loop is broken then that usually results in a doomed timeline.