Historically, the two nuclear devices detonated in war have been set off in the air. So I'll look at those for the effectiveness of bomb shelters.
If you are Ground Zero, you're probably screwed. about 50% of the total power of a nuclear blast is unleashed as a pressure wave. While you are underground and that offers some shelter, the pressure will propagate through the ground as well, and still hit you with enough force to, likely, liquify your organs. A 1MT nuclear device produces an overpressure wave strong enough to inflict heavy lung damage for 3~4 kilometers, not counting the rupture of blood vessels, etc. Again, this will be dampened by being underground, so dial it back a bit, but it's a reference point.
Part 2 of the blast is Thermal Radiation...heat. You can pretty safely ignore this if you are in a bunker, the ground eats up thermal radiation with ease.
Ionizing Radiation makes up about 5% of the blast strength, and this is pure radiation...it moves at the speed of light.
The last 5-10% of the power of the explosion is in residual radiation, which is what we traditionally think of when we think 'Fallout.'
So, as for how your bunker will do against a hypothetical 1MT nuclear device...
If you are within 2.4 km, you are likely dead from overpressure, and your shelter probably collapsed. Out to about 6.2 km, you might survive it, but will probably be bleeding from some orfices and be rather bruised. Beyond that, you might be a bit battered, but should otherwise be okay
You are underground, you can pretty much ignore this.
A properly constructed Fall Out Shelter is designed to reduce the effects of radiation by about 1000x. So, while those on the surface are going to take a lethal dose of radiation out to about 2.3km, and will be quite screwed all the way out to 2.9. You can reduce your radiation exposure by 1000x, since you are in your bunker. So, if you are far enough out to survive the blast wave, the immediate radiation isn't really a threat either.
This stuff gets carried a really long way. On the plus side, it decays exponentially. Roughly 70-80% of the fallout from a nuclear explosion (it is scattered wide, not in big chunks) has decayed within 10-20 days, and it has dropped to reasonably low levels within 100 days. But again, your bunker allows you to reduce radiation exposure by 1000x, so you are still safe, and can probably emerge in less than 100 days, provided that repeated nuclear attacks aren't occurring.
If you are within 19km, you need to be in your bunker when the bomb hits to avoid injury. If you are above ground, the thermal blast will give you first degree burns across any part of you facing the blast (15km = 2nd degree, 12km = 3rd degree, within 10km, you, and everything around you, are now on fire.). You still have time to get below ground before the fallout starts coming down, but you are definitely injured. The thermal radiation is one of the longest range effects of a nuclear blast...being underground when that happens does a LOT to protect you.
Your bunker also needs some more advanced equipment...air recycling and cooling especially. You can't vent to the outdoors, that's full of fallout. And dirt is a great insulator, so you'd overheat within days.
Thankfully, you would probably have notice of an incoming nuclear strike before it hit. We are really good at figuring out missile trajectories, and if an ICBM was headed your way, a warning would go out to seek shelter before the bomb hit...then it's a race.
And aircraft dropped bomb would be harder, as would a suitcase nuke. But, in general, if you are outside of 19km, you could survive this 1Mt explosive without being in your bunker when it hits. Closer and you'd be injured, but might survive, too close, and you are dead meat.
So, all that said, early warning is the best protector against nuclear strikes, if you have a shelter to hide in.