WELL, WELL, WELL...I am sure there are many who have this problem but don't present it in public..but since you did..here I am...
All good answers....always carry tolet paper, but if your a heavy sweater, it will get soggy...so I nornally put it in a plastic baggy...with hand wipes when I have them..(this is something I have learned to do at road races too..)
Drink coffee, another good answer, however, if your in a hurry, sometimes the coffee don't work, and it will make sure you have to go on the run...
I have also used these suggestions to counter the attack of the killer bowels.....
Wear a cheap bandana on your run......one you won't want to wear later....
Never run in heavy populated areas.....
If all else fails, use a sock.....I finished many training runs with only one sock...
How do you battle this problem....I don't have that answer, but since you have an advantage...you know it is coming in one or two miles....plan the early part of your run near a restroom or plenty of trees...
GOOD LUCK...and if you ever have to go on the run and you get to the bushes and you hear somebody say...SHAKING THE TREE HERE BOSS....I already beat you there....