This is more a medical question than a history question. Stroke IS an illness related to both smoking and alcohol abuse. Without knowing too much in depth detail specifically about Churchill’s health, I note that he’d already had some minor strokes (partial paralysis on his lower body) prior to returning to office in his 77th year. Towards the end of his life he had decreased cognitive capabilities, some queried dementia and, of course, a massive stroke took him at 90 years old. These can all be attributed to smoking.

Genetics plays an important part of protecting against things such as high cholesterol, high blood pressure, diabetes, peripheral vascular disease and so on. Environmental factors such as alcohol abuse and smoking tobacco (or repeated inhalation of any noxious substance) also influence one’s risk of developing these diseases/risk factors. Objectively speaking being male, a smoker and an alcoholic puts him into a high risk category for the development of coronary artery disease, atherosclerosis and stroke. I would wager his genetics more than anything helped him live with relatively few debilitating deficits until his death at 90.

The prognosis for strokes in the 1900s (or rather until 1996 when thrombolytics were introduced for the treatment of acute stroke) was poor. The risk of death at 28 days post stroke was 28%, at 5 years post stroke it was 60% (cited below). How Churchill came off so well can potentially be attributed to a healthy diet (the prevalence of highly processed, low nutrition, high caloric food wouldn’t reach mainstream until the 1950s) active lifestyle (in his younger years where ability would allow it), and you can see his robust body in action as it fought off a bout of pneumonia before penicillin was discovered. For an unhealthy man, he was healthy. That comes down to luck (in genetics), and doing pretty much everything else (aforementioned diet and exercise) right.

The cited stroke stat came from: Brønnum-Hansen, H., Davidsen, M. & Thorvaldsen. P. (2001). Long-Term Survival and Cause of Death After Stroke. Stroke, 32 (9). https://doi.org/10.1161/hs0901.094253

In case you’d like to know a bit more about how smoking causes something seemingly unrelated like stroke or heart attack, I’ll elaborate a little. Firstly, a reorientation to the physiology of your body. Vessels in your body allow blood to flow and deliver the necessities to every tissue in your body. A fairly common comparison is a plumbing system (your arteries and veins are like pipes that allow water to flow through the house). Your tissue needs various supplies to function, primarily glucose and oxygen. There’s also other stuff mixed up in your blood - things that can plug any minor leaks (coagulation or clotting, this is why you don’t bleed out when you cut your finger), stuff that identifies and attacks foreign bodies (your immune system and the resulting inflammation) and other stuff which isn’t super relevant, like hormones and so on.

When you inhale a noxious stimulant (like cigarette or cigar smoke, which contains nicotine, carbon monoxide, arsenic etc.) your body does a few things. Firstly, you have the localised inflammation as a result of your body trying to fight off the toxic substances (this can develop into chronic inflammation, and this leads to your common complications of smoking such as emphysema). Secondly, the nicotine actually signals to your body to constrict the vessels (narrow the pipes, essentially) and increase your heart rate (up the flow of water through the pipes), which results in an increase in blood pressure.

Now bear with me. Smoking also alters the way your blood forms clots, making your blood stickier (more likely to clot) and at the same time, damages the lining of your vessels (taking gouges out of the pipes, meaning your blood works harder to patch up the leaks). These two issues may seem to complement each other but they don’t. Once the damage to the lining of your vessels occur, increased turbulence in the vasculature (such as from an increased heart rate and constricted vessels) can cause the plug you’ve built to snap off and travel further down the vascular system. It’ll then get stuck where it can go no further. Some of the most important vessels in your body can be pretty small, such as the vessels which supply the blood to your heart or your brain. If the clot gets lodged there and it gets lodged there hard enough or long enough that the tissue around it starts to die from lack of oxygen, that’s when you experience a heart attack or (one form of) stroke.

One of many reasons why exercise is good for you when it comes to vascular disease is it helps combat the narrowing of the vasculature, makes your heart more efficient and it can create collateral circulation, meaning if your pipe is blocked, there’s another pipe anyway which can supply some or most of the tissues that are no longer getting blood, meaning there’s less tissue death, less severe heart attacks and less mortality.