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100
Just to put some numbers on this, we can assume that burning wood has plenty of particles in the 0.2-0.3 micron range, which is commonly regarded as "the most penetrating particle size".
N95 masks meet the standard that they filter out 95% of these particles. https://www.cdc.gov/niosh/nioshtic-2/20023155.html
In the following study, the better of the surgical masks that was studied allowed 25% of the 0.3 micron particles through, which matches the 1/5th effectiveness you mentioned. https://www.cdc.gov/niosh/nioshtic-2/00210276.html
On p16 of this EPA handout, they absolutely do mention the hand-holding logic of having a false sense of security. But they also address the possibility that the mask can make breathing more difficult, contribute to heat stress, and that these masks:
do not filter out harmful irritant gases, such as acrolein or formaldehyde, or other toxic gases, such as carbon monoxide
They also specifically state:
Masks can also be used in conjunction with other methods of exposure reduction, including staying indoors, reducing activity, and using HEPA air cleaners to reduce overall smoke exposure.
So I think the serious answers to your final questions are: it would help by 25%, I sure as hell would, and maybe?
100
I think an arbitrary kind of mask is effective for COVID-19 largely because of the fluid dynamics: 
If the air has the particles you want to avoid evenly distributed throughout (as with smoke), then this model predicts you'll miss out on most of the benefit of a mask which does the appropriate filtering. So it's probably not worth using surgical masks for smoke.
I'm gonna go with barely, if at all. When you wear a surgical mask and you breath in, a lot of air flows in from the edges, without actually passing through the mask, so the mask doesn't have very good opportunity to filter the air. At least with N95 and N99 mask, you have a seal around your face, and this forces the air through the filter. Your probably better off wearing a wet bandana or towel that's been tied in such a way as to seal around your face, but that might make it hard to breath.
I found this, which suggests that they're generally ineffective. https://www.cdph.ca.gov/Programs/EPO/Pages/Wildfire Pages/N95-Respirators-FAQs.aspx
As should be obvious, this question is somewhat motivated because:
- I'm in Oakland and there's wildfire smoke
- I have asthma
- I have a bunch of surgical masks
- I'm running HEPA filtration but there's enough smoke now that I can still smell smoke in my apartment, so the filtertation isn't keeping up
I've already ordered more dakka, but I'm thinking about things I might do in the interval other than wear my P100 mask, since that has the unfortunate tradeoff that it makes breathing more effortful so it's not a great option for an extended period of time. I'm actually wearing a surgical mask in my home now as an experiment, but I think it's also worth asking since it's a bit hard for me to gather enough data to know if it's making a difference or not.
(The additional context for people who know me in person is that while I've been in Oakland/Berkeley for a the last few years of fires, my asthma is much worse now so I don't know how much the smoke may end up affecting me.)
Sorry to hear about your asthma.
The Powecom KN95s sold here did very well in unofficial government KN95 tests and breathe easier than my P100 respirator:
Adding this improves the seal:
If you look online, you'll see lots of serious people saying that paper surgical masks will not help protect you from wildfire smoke. However, I'm suspicious that this largely reflects a kind of signaling perspective and doesn't consider how effective they might be or under what circumstances and in what ways they might help.
That is, let's say surgical masks are 20% as good as N95 respirators which are 50% as good as P100 respirators at filtering out wildfire smoke (these numbers are 100% fabricated, although I think the ordering is right), then very responsible people will tell you don't wear a surgical mask because it won't help much, and if someone who didn't understand the details just heard "oh, surgical mask is good enough" they might wear one in a situation that lead to them being injured. Compare the lack of nuance in masking guidelines for COVID-19 purposes, and I think you get why I think something like this is going on.
But if a surgical mask was all you had, how much would it help? Enough to be worth, say, wearing one when you otherwise wouldn't wear one in order to help reduce smoke inhalation? Enough to be worth wearing indoors as part of a defense-in-depth strategy combined with HEPA air filtration?