There's no minimum time as evidenced by both the show . Bran is able to go back in time to cause Hodor's mind to go weird, he is also able to see Ned's battle to get to Lyanna. As long as he is alive and/or warging, he can either go to any point in history, or, through different animals (most commonly a raven), he can go to any point in the world (so long as the animal can get to it).
One thing I would like to add though is he can not go into the future. I've seen quite a few youtube videos and questions asking "If Bran can see everything, why can't he do X?"
Bran can only see into the past, not the future
To add on to the above point, the only time we see Bran 'go into the future' is when he touches the special tree and has a vision. AFAIK The vision and warging aren't the same thing If anyone can supply a specific moment where through warging, Bran sees the future, then please do so, but I don't recall at any point in time where Bran went directly into the future through warging.
If Bran can see the very recent past as well, why didn't he tell Theon that Arya was coming? Why let Theon throw his life away?
The first and second part of this sentence are contradictory. Bran can see into the past, however recent, but that specific part you mention of "Why didn't he tell Theon that Arya was coming?"-it's because it's in the future.
Bran doesn't know that Arya is about to save him. He's not expecting Arya to come, he's not expecting anything actually. But he knew that Theon was about to go kill himself. The dude had been through a lot and now the Night King and the WhiteWalkers are literally 10 feet in front of Bran with only Theon in their way. Theon knew this as well so it's either get cut down, or die fighting. Hence the running to his death-he knew he was about to die either way and so did Bran. So it was a (pathetic) last ditch attempt to try and kill the NK (which obviously failed).
But, to answer your question in a summary:
Bran can only see into the past, not the future (the vision wasn't warging), and Arya coming to kill the NK wasn't in the past-that still had yet to happen despite it being seconds/minutes away. So there was no way for Bran to check if he was about to be saved or not.