The short answer is: Sort of. The long answer is: All characters are set at the same point in time, right at the end of the fifth season, shortly after the SJA: Death of the Doctor. That includes the Doctor. However...it wouldn't be the first time the Doctor has tracked over his own time line. It's entirely possible other versions of him might show up...but there must be a plot to justify their arrival.
So...yes. On a case-by-case basis, as we can work them into plot.
❍ Wait, so they're all at the same time? Does that mean I have to play all of the characters set in 2010?
Yes, it does. Most of the time.
❍ But! What if my character happens to be a time traveler, they don't always live their lives in order.
True. And a character that canonically lives their lives in a strange order, like River Song, can arrive at any point and leave and return at a different one, even. But bear in mind they need to have an actual reason and a chance to arrive on Earth in 2010 outside of whatever their established course of events were. Furthermore, if you do bring in a character 'out of order', they will have to maintain the integrity of their time line, so this will be accepted only on a case-by-case basis.
❍ Okay, why should to assume all these people want to work together?
To save the world? For many, that's generally reason enough right there. But for those who aren't so interested in saving the world...the universe, or perhaps even more, no one ever said they had to help. This isn't a jamjar, they've got a life they can live, and the plots of the universe will involve them, even if they're not interested in involving the plots in their lives.
❍ How are they all going to work together?
Well, most of the Doctor's friends tend to be on Earth, and since they're all at the same time frame, it shouldn't be too hard to get some phone numbers and call up some people. Or write them. Or go and see them. And for friends of the Doctor, there's always the possibility of hitching a ride with him in the TARDIS (or on a vortex manipulator, if you know the way).
There may be other ways, as well, as time goes on.
❍ Time isn't going to wibble-wobble on them too much to get this done?
Time is always wibbling and wobbling. But no, it won't wibble so much that you need to worry about people showing up suddenly through cracks in time from the future or the past or knotting up and imploding on itself.
Or...we hope it won't, anyway. There's still that Voice out there who might have different plans.
❍ So if we can app out of any canon, are there any characters we can't play?
Deceased characters whose death would be unaffected by the Doctor's existence or nonexistence. Characters who play minor roles in only one episode cannot be played unless they play as a side-kick to a major one-off character who is present at time of app. Characters which don't have the personality/sentience to be played individually, such as unnamed monsters of the week. In fact, any monster at all will have to be accepted on a case by case basis.
Basically, if your character can't be around Earth in 2010 or the Doctor it's a no. If they shouldn't be but they might be, they could be, but it will be on a strict case-by-case basis, and as plot dictates.
❍ Wait a second, what do you mean 'whose death would be unaffected by the Doctor's existence or nonexistence'? Does that mean there's dead characters we can play?
Yes. In the Big Bang, Rory continued to live the life he lead when the Doctor didn't exist, even though he remembered him. Because River's journal was refilled, that means everything he ever did wasn't undone...but it seems the people whose deaths were a direct result of his interference may just have slipped through the cracks.
Examples of those he 'directly caused' would be companions who died traveling with him, or members of Torchwood (established specifically for dealing with the Doctor) who die in the line of duty. Note that a death by a threat Torchwood deals with not directly related to working for Torchwood (Someone killed by an alien threat because they happened to be in Cardiff, not because they were on a Torchwood mission) would still be dead.
The Doctor isn't responsible for everything, but if you have a 'dead' character you'd like to play who would otherwise be alive in 2010 (or later) not covered by the above two feel free to apply and provide your explanation. It will be considered on a case-by-case basis.
❍ Wait so what about the Time War?
No. That still happened. The Time Lock prevents the War from being affected by the timeline outside of it changing, and Time Lords would be mostly unaffected by a person being erased from time due to a crack anyway.
However...whether a certain rogue Time Lord ended up trapped in the Time Lock later on or not is up for debate.
❍ What about the characters who'll be associating with mine in day-to-day life that aren't currently being played?
Your app will be accepted with the expectation that you can handwave and generally NPC the characters around yours--at least by osmosis--as necessary for interactions. As long as nothing serious happens to or with playable characters around you, you're good.
❍ I have a plot idea, what should I do?
Share it over at the Plot Suggestion board, of course. Or if it's something you'd like to actually do but need approval...share it at the Plot Suggestion and wait for approval.
❍ Right, so when can I submit my app?
Whenever. We have no app cycles.
❍ And can I app for both of my character slots at once?
If you'd like.
❍ Can I use multiple journals for the same character?
Of course, just tell us it's you.
❍ What happens if I drop?
Your character will be relegated to an NPC like all other unplayed characters.
❍ What if I or someone else wants to pick them back up, can I keep the experiences/CR they had before?
Yes, of course, if you'd like to go over the old threads with them and pick up their CR, you're welcome to.
❍ Okay, but what if I don't want to?
Well, those memories that Amy gave everyone else still aren't perfect. Characters might just lose bits, like those weeks or months they spent getting to know someone, and instead believe they'd been living their life uninvolved with everything else.
❍ And what if I want to kill someone?
Well if it's an unplayable NPC, you go right ahead, but know there'll probably be consequences. If it's a playable NPC or PC, even your own, you're going to need to get mod approval for it. This is, after all, Doctor Who. People don't often die in Doctor Who.
❍ But they DO get trapped in parallel universes! Hah!
...Not enough not to need to be approved.
❍ I was rejected? Whyyy? Can I reapp?
We'll probably tell you, actually. And sure you can, but I'd hope you'll have changed whatever problems we had with it by the time you do again. And you'll have to wait a week until you can app again, to give other people a chance.