Hello, I also have eczema but I only get it on my face in cold weather - I find the higher temperature, difference in air pressures and also the sun help improve the skin on my face so I don't get eczema on it, but it seems that doesn't help in your case.
For moisturiser on my face, at the moment I'm using one from Simple with SPF 15 which I quite like. I also like their cleansing lotion, I have used that before but I specifically wanted a facial moisturiser with SPF in it. I have never had my skin aggravated by anything in the Simple range so would recommend them but obviously everyone is different. I also really liked the Magical Moringa moisturiser from Lush when I got a sample but it was very expensive so I didn't purchase it. I also really liked Cetaphil moisturiser but it is a bit pricey for me as I'm on a budget and like I say I wanted moisturiser with SPF. At the moment I'm only moisturising every other morning though on my face as I have acne and my skin is very oily, but in winter I would be doing it probably twice a day to cope with the eczema.
When I have a flare up on my face I use hydrocortisone cream, it is a steroidal cream so you have to be very careful and don't put it anywhere near your eyes. You can buy it OTC but you're not supposed to use it on your face without consultation with your GP really. And the GP can prescribe stronger strengths of it than available OTC but for your face probably a weak strength is better anyway to be on the safe side.
Also I find stress triggers my eczema although not much you can do about it this, as well as lots of toiletries so try to avoid anything that aggravates it (obviously I know that's common sense!).
If you get eczema on your scalp as well or the back of your neck I would recommend trying Head and Shoulders anti-dandruff shampoo and their conditioners, the zinc in it fights eczema, it's the only thing that keeps my eczema on my scalp at bay and also means I can have nice, normal hair and not have to use an expensive product advertised for eczema. When I do get scalp flare ups I move to Oilatum shampoo and if that's not strong enough then I go for Polytar as the next step up, both of which I get on prescription from my GP when necessary.
Sorry for such a long message, unfortunately most of the time with eczema it's just trial and error I think, to see what works for you. Oh also I'm finding the B. products from Superdrug to be very gentle and they don't aggravate my skin if that's another range you want to try, I'm currently using their micellar water and their make-up remover wipes without any problems.