In my experience, watering chillies too much causes them to die. Seedlings are more sensitive, so there's less room for mistakes. If the soil is only slightly damp, then it's perfect. Once the seedlings get to the size you have shown, it's better to water at the bottom of the pot (in the saucer or plate at the bottom, whatever it is called), instead of watering them at the top. The water soaks up, making it just slightly damp, and this helps the roots to grow down looking for water. But not too much water.
Also, rain water is much better than tap water. When I don't have rain, I fill the watering jug or spray bottle from the tap, and let it stand open for a few hours before watering my chillies, so the chlorine etc. the municipality puts into the water evaporates out first.
They also need sunlight for as many hours of the day as possible, and heat. Is it possible to put them outside? It's spring in London now, so not sure if it's still going to snow or frost there, which would be bad, but otherwise, outside it much better than inside.
Another thought - did you transplant them into those pots, or did they germinate there? It's usually necessary to germinate in smaller trays and then transplant, and that's sometimes a risky business. It you mess with the roots too much when transplanting, they don't like it. Try to dig them out with their surrounding soil / seedling mix, so as to keep the roots as they are.