Someone else who grows plants under lights will need to give you advice there. I sometime use supplemental lighting in the spring but that's quite a bit different than having electricity as the sole source of light.
As I understand it, "compost" in England is what Americans would call "potting soil." So, a multi-purpose compost would be of general use for all sorts of potted plants - young and old. Please correct me if I'm wrong.
I have to say that I use a general potting soil for seed starting and right on up . . . It has just always worked for me and I am afraid to risk change. When I once used a brand that was readily available - and, I got lazy - it nearly ruined the entire growing season for me!
Those seedlings are tiny. They certainly haven't advanced beyond the "seed leaf" stage so fertilizing should not be necessary.
What does the bag say is in the mix? It may have a slow-release fertilizer already available to the plants.
I use an organic mix and, without any synthetic fertilizer, it still does quite well for about 4 weeks of growth for my basil. A 5-5-6 fertilizer looks fine to me and basil plants like fertilizer but they would need only a tiny amount. The look of the plants would be a better guide than guessing but there may be a suggestion on the container regarding the amount and frequency of use.