Worlds on the Grand Stair share a ubiquitous present, kept regulated by the Grand Stair itself (in all likelihood). Individual time within a World can be sped up or slowed in some cases - but if left to its own devices, without the interference of a Lord of Gossamer, it will return to regular one second per second linear time.
So - how to fit time travellers, namely the Doctor into such a setting?
The Time Lords are under the mistaken impression they are travelling in time when using a TARDIS, but in reality are directing their Domains to open directly to Worlds of Their Desire.
Time Lords may never navigate the Grand Stair and would only ever encounter a Lord of Gossamer if the Lord visited one of the many Earth-like Worlds that fit within the criteria of the travelling Time Lord.
Note - Time Lords, as an elite group within Gallifrey are a notoriously xenophobic oligarchy. Very few of their number ever leave Gallifrey, let along travel to many different “time periods”. Thus preventing the polution of Worlds with an abundance of Time Lords (and their derivatives from other Worlds that include Gallifrey). The “Blinovitch Limitation Effect” was implemented by an early Time Lord to prevent their fellows from realising they're not travelling in time but going to alternate dimensions - never likely to return to their home dimension.
Time Lords “summoning the Doctor” as occurs in the 3 or 5 Doctor stories is a form of True Name use. With the particular variation that for each regeneration, they vary the True Name, which means they're actually calling a Doctor from a different World.
One of the reasons Worlds are cut off from their Doors is that they have their timeline fiddled with - creating multiple instances of that World. These Worlds detach and drift off into the Void. In the Void are an infinite variety of the same World at different periods in time strung out like an infinitely long necklace on a single timeline without branches.
The space between times of a World on this timeline is filled with a proto-Grand Stair, which the Time Lords have named the “Vortex”. The linking of the World with the one before and after it is very fast, but not instantaneous - thus changing things in the “past” has some lag before the World of “the future” is changed. The amount of time this takes depends on how many World instances are between the changes - an instance of a time is only created when a time traveller goes to that time.
Time travel to different time period Worlds can only be done with the Door-to-Door powers of a Master of the Grand Stair (or a Time Ship such as the TARDIS that has this power)
Time Lords using their Time Ships are limited to traveling up and down within this microcosm of the orphaned World.
In some stories where the Doctor goes to a different dimension - that's when he briefly goes to a “normal” World on the Grand Stair, before going back again to his original Whovian universe microcosm.
A paradox creates a new instance of the timeline and breaks off, forming another microcosm.
Towards the end of the War in Heaven, the Time Lords sought to abandon the universe and find a place to hide from The Enemy. This included the creation and use of a Bottle Universe - a World only accessible from an artifact
There are a number of factions within the Doctor Who universe capable of both time travel and travelling between dimensions. The Time Lords of Gallifrey shun interdimensional travel even moreso than going out and just meeting their neighbours in realtime. However, this doesn't mean that rebellious, renegade or fringe members of the Time Lords, their schisms and splinter groups, have the same inclinations. Mainly I'm referring to the CIA and Faction Paradox. For these organisations to exist, the model of the universe must adhere to Option B
The Celestial Intervention Agency is the only official organisation on Gallifrey tasked with meddling in the outside universe. They behave as an unethical foreign affairs department with a wide range of abuse-of-timelines at their disposal. Primarily because most Gallifreyans regard non-Gallifreyans as animals to be avoided, shunned and put down where necessary. The CIA are likely to attempt the pre-emptive destruction of a race before it evolves (see Genesis of the Daleks). The CIA are also the most likely group to become involved in stopping the anti-authoritarian terrorist acts done by Faction Paradox, and for that reason (and because of the War in Heaven) they meddle in multiple dimensions.
Towards the end of the War in Heaven, the CIA decided to separate from Gallifrey as a separate power, in the naive belief that The Enemy wouldn't destroy them if they were no longer part of their opposition. Manufacturing their own Conceptual Space - a bubble World with specific entrance requirements - called Mictlan. They adopted the name “the Celestis” and exist in Mictlan as Conceptual Entities (in rules terms, every CIA member has Control of Destiny for Mictlan and may shape it as they will). Transforming from physical beings to Conceptual Entities meant that Celesti could no longer enter back into their original universe (the single timeline chain of World microcosm) but must interact with it through agents - those with some Conceptual and some Real parts - and through visitors able to temporarily change into a Conceptual form in Mictlan (a TARDIS can do this - providing non-hostile environments to its passengers in Mictlan)
Although the opposing side stopping the Faction in the book Warlords of Utopia is not named, it can be assumed to be the CIA. In Warlords of Utopia all the alternate dimensions (or Worlds) in which Rome conquered the Earth went to war with all the Worlds where the Nazis conquered the Earth. There are levels of dimension technology implied.
See here for further details - The Faction Paradox Website
Faction members are an example of Eidolon and Umbra Mastery. The time rituals they perform, cutting them off from time etc., bring them more in line with the Grand Stair and Umbra. Whereas the continual creation of paradoxes as an ongoing raison d'être - splintering off microcosms of alternate timeline World chains, or corrupting the original , epitomises the Umbra philosophy.
Faction members are more likely to have personal powers, often invoked with ritual, than to use gadgets.
It should also be pointed out that not only was the Faction founded by someone that never existed - during its history (otherwise known as crimes against history) the Faction has been destroyed or has never existed, or been created on several occasions. If you use the Faction in a game of Lords of Gossamer and Shadow, feel free to completely wipe them out from all time and space - it's been done before.