This is one of those things that you need to mentally practice while you’re out riding by doing “what if” scenarios. In other words, what if I had a flat right now... where would I go and what would I encounter? What if a car pulled out on this steep hill?
To make sure you are mentally practicing the right things, there are a lot of things that need to happen almost at the same time to safely negotiate a stop after a front wheel (or rear wheel) blow-out which actually comprise a “trained response”.
Anyway, for a front wheel blow-out, this is what you need to remember...
a. Pick your riding line (i.e., where you can safely ride to a stop),
b. stay off the front brake,
c. guide the bike in as straight a line as possible with gentle steering inputs down your riding line, and
d. use the rear brake to carefully slow the bike to a stop.
The fine points.
Rear Braking Control: Since you’re trying to minimize the forces on the front wheel that could cause the tire to become unseated from the rim, don't "grab" the rear brake or use aggressive rear braking unless the risk of crashing is less threatening than the risk of going into oncoming traffic, off the edge of a road, etc… Braking with the rear wheel on a bike – and tandems in particular – moves the weight of the bike and rider(s) on to the front wheel as the bike decelerates and the amount of the “load” is directly proportional to how much braking force is being used.
The Front Tire: If you flat a front tire in a hard/fast turn it can get ugly. You have a short window of opportunity to “save it” when the tire first goes flat and is sitting/folded under the inside edge of the bicycle. Keeping the tire on the rim is a function of a firm rim bead on your tire. Therefore, always be sure the tires you use for your front wheel have a good, tight fit on your rims and that they are in relatively good condition with regard to the bead, tread wear and suppleness. A relatively fresh tire is more likely to stay attached to the rim even when flat than one that has a lot of miles or has become brittle from age/ozone/sunlight/dry air.
Front Tire vs Rear Tire Flats: In reality, flatting a front tire happens far less often than flatting a rear, if only due to Murphy’s Law… the rear wheel is technically more challenging to change. Seriously, the heat-induced tandem blow-out is normally something you see with a rear tire since most (but certainly not all) cyclists “ride” their rear brake on long hills and modulate speed with the front brake since it feels and is more responsive than the rear brake. Also, cyclists tend to steer the front tire around (or even pull it up) to clear obstacles that the rear tire gets ridden through or over. Finally, the rear tire carries 60% or more of a bike’s total weight AND is the sole point where power is transmitted from the bicycle rider to the ground which means it wears out more quickly and is more apt to puncture when it contacts a sharp object.