The Tyvek suits don't keep out radiation per se (beta and gamma will go right through) but alpha particles will be stopped, and, more importantly, the contaminated dirt and crud that you pick up on your run to Starbucks after the Bomb drops can be left outside, as opposed to tracking it into your shelter. The real danger is bringing radioisotopes into your shelter, and inhaling/ingesting them.
If there were a "Real" nuke, you'll take the initial hard dose before you can react, unless you're in a real shelter. After the blast, or if it's a dirty bomb/fizzle, the danger is primarily in the aftermath, when the bad stuff is just laying around.
Then disposible coveralls are good.
That said....this show is more of a list of stuff not to do........