We need to divide the problem in half. In the case of an engine we need to decide if it's not starting because of gas or spark. Too test the engine for spark your going to need an extra set of hands or some electrical tools. If you have help you can have some one hold the spark plug wire close to the spark plug while you pull the string. If electricity is getting to the spark plug you will see a spark jump from the wire too the sparkplug. IF you see a spark we can now trouble shoot the fuel.
The easiest way to check the fuel would be to buy a can of ether, that is what the guy in the video sprayed into the engine. If it runs for a couple seconds (or even acts like its about to start) then we know the problem is with the carburetor or fuel system in general.
If you complete these next steps please come back and +1 my answer(so I get a notification) and I will continue to assist you in trouble shooting. Regardless if it's spark on fuel the trouble shooting only get more and more complex from here. I would replace the spark plug and if that doesn't work I would look at calling Home Depot, they might even pick the lawn mover up for you. They are good people.
How to check for spark video https://www.youtube.com/watch?v=mUOmsGiirTU
Aside from environmental factors, you are buying the oil from Vladimir Putin or others fond of AK-47's.
Biodiesel
Heating oil, at least if it's anything like what's used in the States, is quite similar to kerosene/diesel fuel. The difference is diesel is more precisely refined, so it's a little more expensive.
So you can use "green" diesel substitutes, notably, biodiesel.
Most product advertised as biodiesel is in fact nasty, toxic petroleum diesel, with some tiny fraction (2-20%) of actual biodiesel, sold as B2 to B20. The reason is engines need a high-lubricity material like biodiesel to lubricate fuel injection pumps (sulfur used to serve that purpose)... and a fraction of biodiesel accomplishes that.
So in purchasing biodiesel, you must be careful to buy "B100" type, which means 100% biodiesel.
Waste Vegetable Oil
Used fryer oil has four significant differences from heating oil or diesel.
Viscosity (thickness). At room temperature, Vegetable oil is thicker than diesel, which can strain pumps. However, when warmed to about 60C, vegetable oil is thinner and will happily pump like diesel. Further, oil will "freeze solid" at common UK outdoor temperatures. Butter is an example of an oil that has frozen. You cannot pump refrigerated butter. This creates a serious "cold start" problem, if your house were to ever get cold, you would not be able to restart the heater. You can modify the heater to have a method to preheat, or you can modify the fuel by a process called "transesterification" also known as "making biodiesel".
Particlulates (solids): bits of fried potato will clog your fuel injectors. Diesels hate it, but it may be less of a problem on a heater. This is solved by filtering.
water content - this can rust the insides of equipment, and it's a serious concern on diesel vehicles, particularly as the fuel is constantly stirred by motion. It's solved by letting it rest in a still tank, and draining the water out of the bottom.
Acid content - a side effect of frying. This will corrode equipment. It is solved by intentionally adding water, stirring the fuel/water, and titrating (adding lye until the acidity is gone), then, separating out the water. The water is needed to draw out the acidic material, which cannot attach to oil.
There is already a robust business in waste vegetable oil, this is handled by rendering companies who no doubt would sell you waste oil. I gather you want to process it as little as possible. Since you're running a heater and not an engine, you may be able to tolerate a great deal more of the above traits than a diesel car can. If you need to run the full process to remove acidity and thus water, that is most of the work needed to make biodiesel, so you might as well go the rest of the way.
Getting serious: Passive solar design
Using an alternative fuel is just throwing a better carbon at a root problem of atrocious design. Now I get it; you were raised to believe houses should be built the way houses in your area are built. Due to that design, active fuel heating is absolutely necessary.
However, passive solar design can capture a great deal of energy from the sun, and by that, I mean in winter (in summer, the goal is to get rid of energy). This is hard to toss on as an afterthought to a conventionally built house, it works best when you start with the principles and build a new house around it. This can also be augmented with active solar heating, where solar thermal panels collect heat from the sun and store it in a tank of fluid. Again, winter.
Best Answer
Unfortunately the most likely failure point for bad oil is the piston rings. This is a "get a new snowblower" tier problem.
Here's what I would do. Figure out how much fuel you mis-mixed (say 1 gallon). Get a jerry can at least triple that size.
Now, go make some new, additional premix fuel of twice that amount (e.g. 2 gallons) and mix the gas and oil correctly. So you could just use that directly and be good 2 go.
Now, drain the fuel out of the snow blower, and add it to this jerry can. Mix thoroughly.
Now, you have a can of 2-cycle gas that is 2/3 the correct oil and 1/3 not the right oil. The "wrong" oil still has a fair amount of lubricity, and the 2/3 will cover other needs. Go ahead and burn that, white smoke and all.
This won't damage your engine, and is also the simplest way to dispose of this mis-mixed fuel. You certainly shouldn't dump it on the ground, and mustn't dump it down a drain!