> I've found that I really really hate making urgent requests to other teams

For me what ended up working by far the best is to always and for every request try to:

1) Give an honest summary of the negative consequences of your request being delayed/ignored

2) Ask for an honest summary of the negative consequences of your request being honored.

3) Continue calibrating the relative importance until the higher priority task is obvious for both parties.

At first you might get run over a few times but if you don't accept politics (also not your own!) to enter the equation and consistently do this eventually people will be very accepting and these request can be made very efficiently.