I have noticed in my own life and in that of others, that very often during a time of tzarah, Hashem waits until the very very last moment to send a Yeshua. Are there any sources that discuss the reasons why He does so?
The Midrash Tanchuma (Naso:11) brings three things Moshe could not understand; how the Jews can atone for themselves, how the Jew can provide "food" (לחמי) for G-d, and how they can build a Holy Temple where G-ds presence will rest. To all of these things G-d replies:
Which means on one hand that when G-d tests us He only gives us tests that we are able to overcome, but on the other hand these tests are indeed "according to our abilities". (As chassidus explains in several places (For example The Maamar "מים רבים" by the Alter Rebbe or a nice summary of it in this article on Chabad.org), this is in order to bring out our full potential). So since our tests are meant to push us to our personal extreme, therefore to any individual person it makes sense that G-d's help seems to come at the last possible moment.
I just finished reading Rabbi Benjamin Blech's book "If G-d is Good, Why is the World So bad?" While I can't cite to any single page where he makes this point, a major theme of the book is that G-d granted us free will (Deut. 30:15-20). To use his language, if he had not given us free will, and the ability to make our choices, we would be nothing but "hand puppets." But as Deuteronomy 30:15-20 states, G-d has placed on our table choices, good things and bad things, he gave us the tool to make proper choices (the Torah), and he urges us to choose good and life, rather than evil and death. See also, Isaiah 45:7 ("I make peace and create evil").
Based on the above we see that the ability to overcome the evils in this world is within all of us. It is not G-d that has waited until the last minute, but those among us who have just not realized that the power to overcome obstacles is already in their hand. As we will read in this week's Torah portion, Beshalach, this is exactly what G-d told Moses when he and the people were faced with the Egyptian army behind them and the Sea before them. Moses bowed in prayer after telling the people that G-d will fight for them. But G-d tells Moses "Why are you crying to me? Speak to the children of Israel that they should go forward, and lift up your rod and stretch out your hand over the sea and divide it, and the Children of Israel shall go through the midst of the sea on dry ground." Exodus 14:13-16.
Similarly, we learn in Esther that when Israel needs to be saved, G-d has emplaced several people with the capability to deliver us from the evils that would otherwise occur, but maybe for some reason, those individuals have not stood up to accept their role in history. At Esther 4:13-14, Mordechai tells Esther that she has been ideally placed so she can help to deliver the Jews from Haman's plot, and that if she doesn't do it, deliverance will come from soneone else.
G-d has already done His work to save us. The trick is whether we can step up in a timely manner and do what we can to save our people.