This means that there is room to explore the post-Time War universe but prior to ‘Rose’. What are the first things that PCs would do now they are the only Time Lords left in existence.
THE TIME LORDS
The PCs might not immediately give up hope of finding other members of their race. Gallifrey might be destroyed but the Virgin novels suggested there were decoy home worlds in pocket dimensions, maybe there are some to be found there.
Similarly other Time Lords could be exploring other dimensions during the Time War. In your campaign Romana might have remained in E-Space, rather than becoming President of Gallifrey. Maybe the PCs can save her.
They could travel to 2164 and attempt to find Susan. It could be that hiding her under the very noses of the Daleks helped save her from the Time War.
The Master might not be the only survivor who used a Chameleon arch to blend in with local species. The PCs might have valuable information about where to find them and have to work out who the Time Lord in disguise is.
You can either dash their hopes, reinforcing the theme of this era of Doctor Who that they are the only ones left, or let them find other survivors. The reason the Doctor didn’t find them is because the PCs got there first.
With the Time Lords gone does the TARDIS still work the same way? Was the Eye of Harmony destroyed? If it remotely provided power to the PCs TARDIS how will they make their time machine function?
The Doctor’s own TARDIS takes on a radically different appearance, its coral structure revealing more of its organic nature. It also seems to have many improvised replacement parts on the TARDIS console, which makes sense since there is nowhere to replace them.
The trauma of the Time War could have similarly affected the PCs own TARDIS. It could be dying or broken, requiring them to fix it. This could be an excuse to only allow the TARDIS to travel in space but not time or only allow time travel in one direction. If they can only go back in time that would make find new parts even more difficult.
The interior might reveal more of its organic side, as it breaks free from the conditioning imposed upon it by the Time Lords. If the PCs don’t have a good relationship with their time machine now would be a good time to work on how they regard each other, particularly if the TARDIS displays some of the characteristics the Doctor’s ship showed in ‘Journey To The Centre Of The TARDIS’.
Does time travel work the same way? Without the Time Lords who knows what could happen?
Certainly the Time Agents seem to gain more prominence in the absence of the Time Lords. The PCs could find themselves policing the activities of many more species who are gaining time travel technology without the interference of the Time Lords.
The PCs could realise that there needs to be time travel authority. They could be instrumental in picking which race (or races) should run this organisation and what rules they should follow.
THE RESPONSIBILITY OF A TIME LORD
We know that the Time Lords were responsible for much more that just policing time. They seemed to be major players in the universe. The PCs could now be responsible for ensuring allies of Gallifrey continue to be protected and prosper without the Time Lords power to support them.
What of the prisons by the Time Lords? What of the time looped planets? If they survived the Time War what do the PCs do about them? Will they release the prisoners? Eliminate them or let them fend for themselves?
As the last of the Time Lords the PCs might feel their own mortality creeping up on them. They might seek a means to gain a new cycle of regenerations (or just a few more).
Pursuing a technological solution can be dangerous, as ‘Mawdryn Undead’ showed but maybe they can investigate other races who are close to replicating the method. The PCs would then only need to refine their work.
If the PCs are able to find other Time Lords would they do the unspeakable and siphon off a few regenerations (just as the Doctor used his regenerations to save River Song and she did the same for him).
If they follow a dark path the PCs could try to learn the secrets of the Master and steal bodies. To ease their conscience they might only target evil doers, stopping them by taking their bodies. This could lead to complications since they wear the face of an evil person.
It might be that it was the Time Lords that were imposing the restriction on regenerations, allowing the surviving Time Lords to regenerate as much as they want.
THE OTHER RACES
When established races are encountered they should show the scars of the Time War. If they fought in the conflict how did they survive? If they didn’t fight were they forced to evacuate their homeworlds? Did they loose important resources? Were they altered by the Time War?
The events could have changed their whole perspective. This allows you to shift their relationship with the PCs and the role they take in your campaign. Races could be more war-like or become pacifists. Without vital resources their technology could explore other avenues.
If the PCs took part in the Time War they could be held responsible for their actions. Time Lord PCs could be discriminated against purely because of their race. As the last survivor they are expected to pay for the terrible repercussions of the conflict.