We know little of the actual layout of the Time Lord academy. It could be massive or the exterior could be small while the inside is much bigger. I would suggest that it be placed away from the Capital city, in the mountains. This would reinforce the idea that the students are within their own little world.
The academy would be the home of the students and teachers for centuries so would need to be completely self sustaining. In addition to class rooms and lecture halls there would be bedrooms, bathroom, dining halls, kitchens, a medical bay and even a garden for rest and relaxation.
There would be a TARDIS cradle, where time capsules could be kept for field trips. Students could be assigned to care for the TARDIS’s as one would a school pet. As living creatures someone would need to tend to them and this would give player characters an opportunity to bond with a TARDIS and even slip away for a short while.
The library would have access to a vast store of information, stretching back centuries. Not only would this aide a student studying but it could contain secrets that can spark an adventure.
It might also make a tempting target for enemies of Gallifrey. With only minimal security and surrounded by mere students and dusty teachers obtaining the secrets of the Time Lords might be much easier at the academy. Unless the player characters can stop them.
Zero rooms can be a place of quiet contemplation, useful when the stresses of exams get to much, or a place to heal and recuperate. Reaching the Zero room could be a goal for the player characters if one of their number has been critically injured.
Student and teachers living quarters would off course be kept separate. Further more students are also divided into chapters. The ones that we know are Arcalian, Cerulean, Dromeian, Patrex, Prydonian and Scendle.
All of these chapters will be explored later, as we look at the culture of the academy. A games master can introduce as many chapters as they like. They are important in that they give the students something further to attach their identity to.
Students in the same chapter will spend more time with each other, strengthening their bond. Chapters relationships with each other can be an important factor in how encounters turn out.
Chapters with the same view point will get along better than those who are rivals or enemies. A player character might be an enemy with a student in his chapter but if they come into conflict with another chapter they are likely to put aside their differences to stand against a common enemy.
If you decide that assignment to various chapters is based on personality then it can be a good short hand to describe NPCs. By telling the players what chapter a student belongs to they will quickly know what kind of person they are.
‘Lungbarrow’ introduced the idea of avatroids, domestic robots with a bestial exterior. It acted as the Doctor’s protector and nanny. It isn’t a stretch to imagine that such robots would be put to use in the academy, doing menial tasks such as cooking and cleaning.
These avatroids are programmed with basic emotions and personalities. This allows them to be characters in their own right, giving player characters a chance to develop a relationship with them.
To get from and to the academy students might use a transmat system or air cars. If you need to isolate the academy these systems might malfunction or be sabotaged, cutting the students off.
There would be forbidden areas that would hold their own mystique. After all, what is more tantalising than a place you’re not supposed to go?
The Untempered Schism might be located within the academy, or at least close by. A gap in the fabric of reality looking into is a rite of passage for the students. The experience changes them forever.
Who knows what affect repeated viewing might have? Its very existence is a mystery and students might try to prove themselves by learning how it came to exist. Students might even challenge each other, proving their bravery by looking into the schism once again.
The school might also have access to the Death Zone. Its reputation is fearsome enough to keep most students away but they might have cause to go there anyway. They might be searching for the remains of the alien who died there to help them complete a project or it could be the perfect place to hold a secret meeting.
Similarly enemies of Gallifrey might lurk there, getting ready to strike. If the Death Zone is particularly close a malfunctioning transmat or air car crash might strand player characters or someone they know there, requiring a rescue mission.
We know that Gallifrey went through Dark Age and that texts such as the black scrolls existed. If the academy has stood since those times there might be concealed areas with hidden treasure and forbidden knowledge to tempt those with dark hearts.
The Doctor Who universe doesn’t support the existence of ghosts but a hint of the supernatural can be introduced with temporal echoes. With the presence of so much arton energy it is easy to imagine that over centuries the images of past residents of the school might bleed through.
These spectres of the past might reveal hidden secrets of the academy to students. They might even see temporal echoes of the future, giving them prophetic visions of what is to occur at the academy.
The academy is where the player characters will spend most of their time so it should feel familiar. They should know what is there and have to get there. The fact that they share it with so many people ensures that it won’t feel intimate but it will be their home.
Over the course of the campaign you can reveal new areas or have parts destroyed, rebuilt or expanded. This will ensure that the setting feels fresh, even after the player characters have spent years there.
When they leave they should feel sadness and pangs of nostalgia should they ever return. To do this have memorable events be tied to specific locations. They should remember the argument they had with their arch rival on the great stairs or searching for the missing amulet of Omega amongst the avatriods in the academy kitchens or the first time they saw a TARDIS in it’s cradle.
Not only should the academy be where the player characters operate from but a source of adventures in itself.