I can recommend Filetti, Fairy or Simply. Filetti can be a bit more difficult to get your hands on depending on where you're living.
Fairy (P&G) and Persil (Unilever) are the most readily available and well advertised laundry detergents for babies and children. Fairy's only sold as non-bio, but most non-bio detergents are good too. In some countries P&G sometimes switch between Fairy and Dreft for washing-up liquids, so it could stand to reason the same could be happening at times with laundry detergent.
If you want something a bit more specific and a little less commercial range I can recommend
Simply. It's designed specifically for sensitive skin (supported by national Eczema Society), Eco friendly and Vegan. It's currently on offer 2 for £5 at Sainsbury's.
ETA: Don't forget laundry detergents are going to perform slightly differently depending on what area of the country you live in and what your water hardness is.
But, it shouldn't be anything too drastic. (You just need to use a higher dosage of detergent in hard water areas to maintain cleaning performance and avoid the build-up of dirt and hard water deposits in your machine.)