I have some suggestions for alternative emulsifiers, and a concern to express using eggs given the intended purpose of your syrup.
As to how "every-day" these are, it will depend on where you are in the UK. I can find these easily in the states at places like Whole Foods, Trader Joe's, etc in the bulk spice/flour aisle. Lecithin by itself is a lovely emulsifier, and usually readily available in candy/cake decorating shops. Here are two more:
- Gum Arabic
Bartenders have been making a "gomme syrup" as the simple syrup base for classic cocktails for many-many years. The gum arabic provides a really silky-velvety mouthfeel without being too viscous. It is a 1:3:6 parts gum:water:sugar ratio. Recipe Ex: 1 gram gum arabic, 3 grams water, 6 grams sugar(fun twist...try using invert sugar, honey, or malt extract for extra flavor depth). The sugar and water will be heated together until they have dissolved, followed by the addition of the gum, and vigorous stirring. Hydrating part of the gum in a bit of your water allotment overnight prior to making gomme syrup is very helpful. You can choose to add the ginger/lime/alcohol based flavorings at the end of this step, or while the emulsion is being created.
After making your gomme syrup, create your emulsion by adding the almond oil to the syrup slowly while you blend with an immersion blender (or some other high speed mixing device). If you have any other flavorings to incorporate, add them at this step.
These are all thermoreversible hydrocolloid gelling agents that when hydrated will help provide emulsifying power for you. Guar gum, particularly, does better if it has had 3+ hours to hydrate before you heat it...from personal experience. Amounts of these will vary, but there are some really great synergistic benefits you get from combining xanthan and guar gums(increased holding power, etc). Experimentation will give you the best final ratios.
Note on Eggs for Cocktail Syrup Use
I love eggs in cocktails. Most of the experience I've had with their addition is usually at the time of the drink preparation, specifically for its immediate service. Yahtzee.
However, making a cocktail syrup that you will be keeping around for a while with raw eggs gives me pause in this case. Unless you have a way to pasteurize the eggs or the final mixture, you may wind up harboring a rancid mess with quickness.