"I'll arrive a mess"
Here's a belief that's mostly about a picture. You imagine arriving at the office, or the meeting, or the school gate, red-faced and damp from the walk, and the picture is enough to put the keys in your hand.
A mile at three miles an hour is a walk you can talk through. That's the NHS's own definition of the pace, talk but not sing, and it's chosen because it's the pace at which you get the benefit without the sweat. A person who's warm from twenty minutes of walking isn't a person who's been running. They're a person with a bit of colour, and the colour goes in the five minutes it takes to hang up a coat and sit down.
The NHS walking page has a plain answer to the clothes question, which is to wear your usual work clothes with a comfortable pair of shoes and change shoes when you get in. That's the whole of the logistics. A pair of shoes under the desk.
Rain is chapter 10. Wind is a coat and a minute in the loo with a comb. Hills are chapter 22, and if your walk to work has one, walk it a little slower and arrive a minute later and no warmer.
There's a small subset of readers for whom this is real, the ones whose walk to work is two miles with a hill at the end in July. For them the answer is a shirt in the bag and two minutes in the bathroom, which is what cyclists have been doing for a century, and which stops being a thing to think about after the first week. For everyone else, the mile at the talking pace doesn't produce anything a coat and a comb don't handle.
The thing to notice about this belief is that it's a forecast. You haven't arrived a mess, because you haven't walked. You've imagined arriving a mess, and the imagining has a red face in it that the walk wouldn't produce. Walk it once, at the talking pace, with the shoes under the desk, and see what you look like in the lift lobby mirror. You'll look like someone who came in from outside, which is what you are every morning anyway.