I can't guarantee it's 100% accurate, as they do spout technobabble rather fast, also I couldn't have it on very loud as my housemate's asleep, but I did my best.
Edit: Watched it again with the volume cranked - I think all the gaps are filled in now.
Martha walks to the tardis door
MARTHA: I'll see you again, mister.
She exits the tardis, pauses, then walks away. Inside, the (tenth) Doctor leans against the console, looks thoughtful, then operates the controls.
The tardis spins out of control and a klaxon sounds. The Doctor struggles back to the controls.
TEN: Stop that! Stop it! What was all that about, eh? (He taps the time rotor) Eh? What's your problem?
As he goes to investigate the controls, the fifth Doctor appears on the other side of the console.
FIVE: Right, just settle down, now...
Going in opposite directions around the console, they bump into each other.
TEN: Excuse me..
FIVE: So sorry...
They stop and look at other.
FIVE: Who are you?
TEN: Aw, brilliant! I mean, totally wrong, big emergency, universe goes bang in five minutes, but... brilliant!
FIVE: (angrily) I'm the Doctor, who are you?
TEN: (chuffed) Yes, you are, you are the Doctor.
FIVE: (exasperated) Yes, I am, I'm the Doctor.
TEN: (Still chuffed) Oh, good for you, Doctor. Good for brilliant old you.
FIVE: Is there something wrong with you?
TEN: Ooo, there it goes, the frowny face! I remember that one! Mind you, (he grabs the fifth Doctor by the face and squishes his cheeks) it's saggier than it ought to be, (he fiddles with Five's sideburns) hair's a bit greyer, that's cos of me, though, two of us together has shorted out the time differential, should all snap back in place when we get you home, (he grabs him by the lapels) be able to close that coat again. But never mind that, look at you! The hat, the coat, the crickety cricket stuff, the... stick of celery, yeah... Brave choice, celery, but fair play to you, not a lot of men can carry off a decorative vegetable.
FIVE: Shut! Up! (Snatches off his hat in anger) There is something very wrong with my tardis, and I've got to do something about it very very quickly, and it would help, it really would help, if there wasn't some skinny idiot ranting in my face about every single thing that happens to be in front of him!
TEN: Oh, okay. Um, sorry. Doctor.
FIVE: Thank you. (He turns back to the console)
TEN: (Hugely enthusiastically) Aw, the back of my head!
TEN: Sorry, sorry, it's not something you see every day, is it, the back of your own head. Mind you, I can see why you wear a hat... I don't want to seem vain, but could you keep that on?
FIVE: (Turns to face him) What have you done to my tardis? You've changed the desktop theme, haven't you? What's this one? Coral? It's worse than the leopardskin. (He put on a pair of half-moon spectacles, and turns back to the console)
TEN: (bouncing) Aw, and out they come! The brainy specs! You don't even need them! You just think they make you look a bit clever!
A klaxon sounds.
FIVE: That's an alert... Level five. Indicating a temporal collision! It's like... two tardises have merged, but there's definitely only one tardis present... (As he rushes around pressing buttons, the tenth Doctor strolls across, leans on the viewscreen and watches him) It's like two time zones at war in the heart of the tardis... That's a paradox. Could blow a hole in the space-time continuum the size of... (Ten shoves the viewscreen across into his eyeline) Well, actually, the exact size of... Belgium. That's a bit undramatic, isn't it? Belgium?
TEN: (Pulls the sonic screwdriver out of his pocket) Need this?
FIVE: Nah, I'm fine.
TEN: Oh no, of course, (does a rather swish flipping and catching move, and puts the screwdriver back in his pocket) you mostly went hands free, didn't you, like 'eh, I'm the Doctor, I can save the universe using a kettle and some string, and look at me, I'm wearing a vegetable'.
FIVE: (Stalks over to stand nose to nose) Who are you?
TEN: (Softly) Take a look.
FIVE: No. Oh, no...
TEN: Oh, yes.
FIVE: You're... oh, no...
TEN: Here it comes... yeah, yeah, I am...
FIVE: (With disgust) A fan.
TEN: Yeah... What?
Something on the console beeps.
FIVE: Level ten, now. This is bad. Two minutes to Belgium!
TEN: What'd'ya mean, a fan? I'm not just a fan, I'm you!
FIVE: Okay, you're my biggest fan. Look, it's perfectly understandable, I go zooming around space and time, saving planets, fighting monsters, and being, well, let's be honest, pretty sort of marvellous... (Ten nods in agreement) So naturally, now and then, people notice me. Start up their little groups. That LINDA lot. Are you one of them? How did you get in here? Can't have you lot knowing where I live...
TEN: Listen to me, I'm you. I'm you with a new face. (Slaps his own cheeks) Check out this bone structure, Doctor, cos one day you're gonna be shaving it.
A bell sounds, causing the Who purists in the room who've been making snarky comments during the earlier klaxons to feel a bit silly.
FIVE: The cloister bell!
TEN: Yep, right on time. That's my cue...
FIVE: In this limit, we're gonna generate a black hole strong enough to swallow the entire universe!
TEN: Yeah... That's my fault, actually. I was rebuilding the tardis, forgot to put the shields back up. Your tardis and my tardis... well, the same tardis, different voyages in the same time stream, collided and wurp, there ya go, end of the universe, butterfingers, but, don't worry, I know exactly how this all works out, watch... (he fiddles with the console) Venting the thermo buffer... Flooring the helmic regulator... And just to finish off, let's fire those zyton crystals.
FIVE: (Pulling Ten's hands away from the controls) You'll blow up the tardis!
TEN: Only way out.
FIVE: Who told you that?
TEN: You told me that! (He hits the controls, and the tardis whirls through the time stream. Everything fades to white, then returns...)
FIVE: A supernova and a black hole at the exact same instant...
TEN: Explosion cancels out implosion.
FIVE: Matter remains constant.
FIVE: Far too brilliant. I've never met anyone else who could fly the tardis like that.
TEN: Sorry, mate, you still haven't.
FIVE: You didn't have time to work all that out. Even I couldn't do it!
TEN: I didn't work it out. I didn't have to.
FIVE: You remembered.
TEN: Because you will remember.
FIVE: You remembered being me, watching you doing that... You only knew what to do because I saw you do it.
TEN: Wibbly wobbley...
BOTH: Timey wimey!
(The tenth Doctor raises his hand for a high-five, but there aren't any takers. Another alarm sounds and he leaps for the console)
TEN: Right! Tardises are separating. Sorry Doctor, time's up, back to long ago. Where are you now? Nyssa and Tegan? Cybermen and Mara and Time Lords in funny hats and the Master? Oh, he just showed up again, same as ever.
FIVE: Oh, no, really? Does he still have that rubbish beard?
TEN: No, no beard this time. Well, a wife.
FIVE: (Beginning to fade) Oh. I seem to be off. What can I say? Thank you. Doctor.
TEN: Thank you.
FIVE: I'm very welcome. (Disappears)
(The tenth Doctor flips a switch on the console and he reappears)
TEN: (Picking up Five's hat which was left on the console) You know, (hands him the hat) I loved being you. Back when I first started at the very beginning, I was always trying to be old and grumpy and important, like you do when you're young. And then I was you. I was all bashing about and playing cricket and my voice going all squeaky when I shouted, I still do that! The voice thing, I got that from you! (The fifth Doctor smiles and puts on the hat) Oh! (Puts his foot up on the console) And the trainers! And... (Pulls a pair of glasses from his pocket and puts them on) Snap! Cos you know what, Doctor? You were my Doctor.
FIVE: (Raises his hat) To days to come.
TEN: All my love to long ago.
(The fifth Doctor disappears. Ten smiles at where he was)
FIVE: ('s voice) Oh, Doctor? Remember to put your shields up.
(The tenth Doctor hits a button on the console, and immediately afterwards, the Titanic smashes through the wall)
TEN: What? What? What?