‘Time Heist’ introduces Saibra, a shape-shifting mutant. The existence of mutants allows you to mix Doctor Who and X-Men into a single setting. Mutations are a good way to give human characters interesting powers.
There are many different ways in which these mutations might occur. It could be a natural evolution, with humanity reaching a stage where these powers manifest in new generations with increasing frequency.
This stage need not be restricted to the far future. They could occur throughout history. Certain periods could have a ‘boom’ of mutations which might eventually reduce if their blood lines don’t continue. PCs might be part of this current boom, their powers bringing them together. This could allow you to link the Doctor Who universe to the setting of the Tomorrow People.
There could be outside factors that artificially create mutations. This could be a side effect of designer genes, futuristic medicine or the result of radiation from spaceship engines or living on alien worlds. The question would be whether anyone is held accountable for these genetic mutations.
Interbreeding with aliens might also result in children inheriting abilities. Alien genes might be recessive, only activating several generations later. The mutant might not be aware of their inhuman heritage.
You could have several explanations for the mutations. People might not know why they are different, it only matters that they are. How they react to this will depend on the society they live in.
Mutants might be treated with fear and suspicion. Saibra certainly doesn’t see her powers as a gift and this might stem from a society which views her as an aberration. This kind of setting can serve as the backdrop for an adventure that explores tolerance.
Other societies might revere mutants and envy their powers. Mutants could form the elite of society, maybe even the ruling class. Mutants could use their powers to help and protect others, making them similar to superheroes.
Of course an open minded society might not treat mutants differently at all. They could just be an accepted part of society. Such a society could seem ideal for those mutants who live under oppression.
Mutants and how they treated could be an important political and religious issue. It could ally some galactic communities or ignite war. Changing opinions can help shape the galactic map over the years.
How a society reacts to mutants could depend on what powers a mutant can manifest. They could be low key (able to change hair colour at will, be triple jointed, an extra finger, etc), monstrous (natural armour, the ability to spit acid, burning touch, etc) or godlike (invulnerability, super strength, control weather, etc).
Mutant gifts often have a tragic element to them. Saibra can shape shift but can’t control it. She considers herself forever alone because she assumes the form of anyone she tries to get close to.
If you wanted to keep this theme there should always be a downside to the mutant power. It could be that a mutant can fly but their bones are also much lighter, making them brittle. Their super speed or strength could require them to replace their lost energy by eating large amounts of food. With PCs this can help balance their extraordinary abilities and for enemy NPCs it can provide a weakness for the PCs to exploit.
A Time Lord PC could take it upon himself to rescue and protect mutants. This could be because he is one himself or maybe he is directly or indirectly responsible for creating mutants.
This can be a good excuse to take the Time Lord to a variety of planets, as he hunts down rumours of mutants. Along the way he can challenge prejudice, with those he takes under his wing serving as his companions. Each new mutants abilities could be the source of further adventures as they learn to use them.
UNIT and Torchwood could encounter mutants during their investigations. It is easy to imagine that their unusual abilities might at first be attributed to aliens. Once the organisations realise what they are and that more exist how would they react?
If you wanted to make mutants a hostile force then they could all be driven to replace humanity. It could be that mutant powers are always linked to some form of psychotic or homicidal behaviour. Evil mutants could be rogue monsters or a secret society living amongst normal humans while plotting their downfall.
Mutants could be powerful allies, especially in the face of alien invasion. With extraordinary abilities and powerful natural weapons they could turn the tide in humanities favour.
UNIT might organise a special mutant squad to deal with their most difficult missions. An adventure set during the Dalek invasion of Earth could see survivors seeking out mutants amongst the rubble of London to repel the invaders.
Other species might have mutations of their own. The Daleks, with their desire for genetic purity, would probably exterminate any mutants but some might be able to hide their powers. Their mutation might also manifest as a way to think different, opening the possibility of a ‘good’ Dalek.
Mondasian Cybermen, if they are genetically similar to humans, could develop their own mutants. They might also convert human mutants, strengthening their ranks with their gifts.
As a clone race a mutant Sontaran could result in his entire clone batch being eliminated. If a mutant Sontaran was to prove his worth in battle his DNA would likely be used as the source for subsequent clones, resulting in more mutants with similar powers.
Their identity as mutants could allow different species to come together. This could result in an evil mutant empire or a progressive society that shows that universal peace is possible.