My go-to place for quick rules summaries are the reviews on RPG.net, and in this case it doesn't disappoint: there are eight reviews of Doctor Who: Adventures in Time and Space on RPG.net. Picking one review at random (well, if "click the first link" is random…), it has a good introduction to the core mechanic with an explanation of how it hangs together and an example, though it's not an example of real play:
The rules themselves are simple and quite reminiscent of the Cinematic Unisystem--roll 2d6, adding Attribute and Skill, and attempt to get a target number (generally 12 or better). There are levels of success, and in a neat twist, levels of failure. Rather than having a flat score representing a level of success (the aforementioned Unisystem, for example, has 9-11 being one success, 12-15 two, etc.), this game bases levels of success as a range above or below the target number (which can slide up or down from 12 based on difficulty.
If it sounds complicated, it's not. Let's say Rose is trying to leap over a pit. It's a long pit, longer than a normal person could pull off. But she was a Gymnastics champion, so she'll give it a shot. The GM decides that while a normal difficulty would be 12, this is a tricky task--that sets the difficulty at 15 (there is a table with difficulty guidelines in the book). Rose rolls 2d6 and aces a twelve! Adding her athletics skill and coordination attribute to it gives her more than enough to make the leap. Depending on how high above the difficulty she makes it, she could just barely succeed, meaning perhaps she just hits the edge, her foot slips, and she finds her self not falling, but hanging from the ledge and must try to pull herself up. Or she could pull it off just perfect, or she could hit the ground, roll into a somersault, and come up running before her pursuers have a chance to react.
The nice thing about this is that everything you do in this game works off of exactly the same principle. There are no subsystems, no efforts to cram obscure possibilities into the rules […]
The review goes on to talk about the only other core mechanic, Story Points, and how they're pretty much exactly what most readers will expect as the familiar Fate/Hero/Author point that lets you add a useful detail or turn a hit into a near miss, etc.