Is there a more environmentally-friendly alternative to using oil to heat a house with oil heating installed?
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.