Another thing I need people to stop just saying like it's the truth is that Bruce picked up Tim as Robin immediately after Jason died, or that Tim forced his way into being Robin.
Now, of course, there is the caveat that Jason can come back into town thinking that it was the case, he was dead and Talia informed him of the changes in Gotham, so we don't know how accurately he was informed. Another caveat is that Tim can grow up thinking he forced his way into it because he does argue for it at a point, but I need no one else who had a fully developed mind and was there saying it. Okay? Okay.
So, let's tackle the first one. Jason Todd gets an official death certificate in the Daedalus and Icarus Annual explaining how he came back showing his date of death as 4/27/19xx (since comics are always meant to be contemporary unless stated otherwise, we can just ignore the year). The DC Encyclopedia had a death certificate too, but it had a lot of odd details, not least his height and weight being more in line for a 10 year old than a 15 year old (the Titans comic put his height at 5'4" when he died), so I'm going to use the comics certificates over that when they conflict, but the Batman Annual also lists his date of death as 4/27 (why is it a Gotham City Death Certificate when he died in Ethiopia? I have no clue).
(Batman Annual 25)
(New Titans Vol 1 55)
(DC Encyclopedia)
Tim Drake appears after Jason dies when Bruce Wayne is going a little crazy as Batman. He specifically shows up during a break week from school.
(Batman 1940 #441)
Given that he could only have a Break Week feasibly in November for teacher conferences, December for Christmas/New Year's, or March/April for Easter, and the greenery outside show that it's neither November nor December, he's therefore shown up for his Spring Break. Since that would never be after April 27th, then he has to be there at least the first Spring Break following Jason's death, so about a year later.
He shows up and begs Dick to become Robin again to save Bruce from himself.
(Batman 1940 #441)
You see, this is interesting. He's explaining all his best qualities, not as a way to say "I'm right for the job" but to awkwardly (as only Tim Drake can) explain how much Batman and Robin mean to him, and how much they've influenced his life, to show how committed he is to this idea to back-up his assessment that Batman needs Dick as Robin again.
Dick is very adamant in his refusal to step back into the Robin role.
(Batman 1940 #441)
Both Alfred and Dick(Maybe? unless it's just Alfred being the drama again, but Dick did leave his old Robin costume out on the side before leaving) suggest Tim should pick up the costume. And he does only hours later, when Dick and Batman's signals have disappeared after going to confront Two-Face. Tim is feeling guilt over Nightwing possibly being dead because he got involved, he's worried sick about his heroes, and Tim has to go help because he is not built for sitting back and worrying.
(Batman 1940 #442)
At this point, he does start trying to convince both himself and Batman that he should be Robin, but only seriously after Alfred and Dick start hyping Tim up as well. Bruce is not having it. Like. At all.
(Batman 1940 #442)
They go to track down Two-Face and the confrontation ends with Tim nearly getting crushed to death as a pile of cars in the junkyard are toppled over onto the Batmobile, where he's been instructed to wait.
(Batman 1940 #442)
The confrontation ends with Bruce punching Two-Face and telling him that he can't kill Batman or Nightwing (Or Robin?) Or Robin!
Afterwards, after all the adrenaline has worn off and they're back in the manor in their normal clothes, Tim starts to back off of being Robin. He's still super stoked for having had the chance to fight criminals with his heroes, he's 13, give him a break. Please and thank you.
At this point it is Alfred, Dick, and Bruce who cement Tim going into training to become Robin. Yes, because of his logic, but also, like. I think Bruce knows deep down he needs someone to work with just as much as everyone else does. And Tim already knows his secret, and this is the easiest way to keep him quiet. Also, Bruce. Fucking. Communicate better. Jesus.
(Batman 1940 #442)
And then Tim's training begins, but he won't become Robin until the following Christmas. And only after he saves Bruce because he's been captured by Scarecrow and is being tortured with fear toxin with Vikki Vale. Tim was ordered to stay home, but again, he is incapable to standing back and not helping when he thinks there's a problem (this is a good and heroic character trait). This time, he doesn't wear the costume because he can't bear the idea of failing in it, so he dons a ski mask and comes in, gets dosed with fear toxin while saving Batman and beating Scarecrow and runs away before the cops show. Tim believes this means he'll never be Robin, that Bruce will fire him from training, but at least Bruce is alive. When Tim explains this all to Bruce, Bruce says the rules are there for a reason, but sometimes it's okay to break them when justified and gives Tim a new Robin costume.