Strip off.
Your skin is waterproof.
Assuming it's warm (because if it wasn't, you'd have proper gear, right?)
then you're not going to freeze, especially if you're walking.
So take off everything you can afford to be seen without (because getting arrested for flashing is a downer) and stash it in your pack.
When the sun comes out again, you can air dry as you go, or you can sacrifice some item of clothing as an impromptu towel.
I've done substantial amounts of summer walking in the UK wearing only a pair of shorts, most notably on the West Highland Way in Scotland where it rained sideways 4 days out of 6.
If your work rate is high enough, you stay warm from the inside in spite of the water running off your skin.
outdoors.stackexchange.com/users/6481/Graham
So, if you're caught off-guard, you won't have anything with you that offers direct protection, such as: raincoat, poncho, garbage bag, tarp, or any other portable shelter.
It also means that you won't have any tools along that might help you build a shelter, such as a pocket knife.
Some might have one, some not, but let's assume you don't.
Some of these options will reduce your wetness, but not eliminate it.
Some of these are not quick solutions and wouldn't be fast enough to help for a quick downpour.
Therefore, you'll need to quickly consider your environment, and look for what is available.
The possibilities will differ greatly: City/Suburbs: alcove of a building, or hopefully you can go inside
(you're probably not thinking of this environment, but for completeness) if there is a building, find the wall which is most out of the wind.
Stand against the wall, as flat against the wall as possible, and you will be out of the worst of the rain.
It will work with big trees as well, but trees mostly have branches which will either help or hinder (drops.)
Forest: under an evergreen of some kind.
If there's lightning, be careful here.
Definitely avoid the taller trees in that case, but that may not be enough.
a cave would be great a cliff might help, either because it's overhanging, or because the wind is blowing the right direction gather dead branches to build a quick wickiup, and cover it with leaves if it's a genuine survival situation, use your hands to break boughs from evergreens and make a shelter There will be environments where there's no good solution if you're caught off guard.
These might include open plains, desert, etc.
outdoors.stackexchange.com/users/1942/Don Branson
My experience as a cyclist is that getting wet doesn't matter (because skin is waterproof): what matters is getting cold.
So the goal is to stay warm enough.
Your warmth is affected by your exercise (producing heat) and insulation/clothing (retaining heat).
To keep warm, keep exercising: if you stop and take shelter (when already soaking) then you'll get cold[er].
It may be better to keep on keeping on (even in the wet) until you reach an indoor shelter.
In dry weather you might keep your jacket unzipped (to stay cool and avoid sweating); if you're drenched in rain, staying cool and sweat-free is less important, instead you might zip up to keep warm.
outdoors.stackexchange.com/users/6365/ChrisW
Get out.
You can quickly cool down when it's raining out, in particular when the rain is horizontal and it's 4°C outside.
You say there's no shelter anywhere near.
Then walk as fast as you can to somewhere not near.
I infer yours must be a day trip, for
surely you're not out on a multi-night backpacking trip in a cold & wet climate without any raingear or night shelter.
Locate the closest shelter,
be it a bus stop, a pub, your car, or even a house if you trust strangers and they trust you.
Even if they don't invite you in, you might hide under a ledge.
Walk to such shelter as fast as you reasonably can, be it one, two, or three hours march.
The wet-cold spiral is real and bad for you.
P.S.
It amazes me how often I see people in England out in the rain without proper rain protection.
This is England, it rains.
I don't recall seeing the same in Netherlands, Sweden, or Norway.
In the USA too, right on the peak of Mt. Marcy, I remember seeing hikers without any rain gear caught out in heavy rain.
A sure way to catch a cold or worse.
outdoors.stackexchange.com/users/566/gerrit
A waterproof map made of a rubber-like paper.
I keep one in my backpack for going to the library.
It is very small and light-weight when folded and very big when opened up.
This is much more durable than a garbage bag.
Trails Illustrated have them.
They can also serve as something to sit on if really need to do so, and the area is wet.
They dry fast, as well.
outdoors.stackexchange.com/users/14756/Planet.Megan
If you are caught out in the rain without any rainwear, raingear and nowhere to hide out, how can you minimalize the damage?  I've seen things like:   holding a bag over the head  wearing a plastic bag on the head  keeping the head down   Obviously planning ahead would be ideal, but...:  What methods are there to avoid general wetness when you're caught off guard.
Caught in the rain without rainwear or raingear
outdoors.stackexchange.com
<walking><rain><technique>
outdoors.stackexchange.com/questions/19535
outdoors.stackexchange.com/users/6665/Mo.
Getting wet doesn't matter (because skin is waterproof). What matters is getting cold - so the goal is to stay warm. Strip offf if necessary. Alternatively, you could use a waterproof map made of a rubber-like paper. Waterproof maps can be handy. Skin is waterproof and you should focus on staying warm.
Skin is waterproof, focus on staying warm. Waterproof maps are useful.