My sister went through a bad bout of this as well-- hers was actually due to an obsessive-compulsion to wash her hands after touching anything that might have bacteria on it. (And really, what doesn't?) Luckily this compulsion diminished with anxiety control, but her hands were a mess. She tried plenty of products as well that make claims they can't live up to. What worked for her was coating her hands with Vaseline before bed. She even tried slathering her hands in it and then putting on rubber gloves, which she said felt a little odd at first, but once or twice a week helped a lot. During the day, she would keep a heavy hand lotion with her like Neutrogena Intensive Care and apply it after washing. The biggest offender is alcohol-based hand wash, which is unfortunately used most in hospitals and clinics for its anti-baterial properties. But there are gentler options containing soothing ingredients that you might want to invest in yourself and bring along to work.
Aloe vera is excellent for preventing peeling, but it is tough to find the pure gel nowadays, which is why some who say they're using aloe vera gel aren't satisfied. Your best bet is to buy the plant or check your local health food store for all-natural aloe vera creams.
Peeling and flaking hands can be due to contact dermatitis and contact with harsh detergents, soaps and lotions. Sometimes cold, dry air can cause peeling and dermatitis or fungal infections can also cause peeling.
Wash the areas several times with fresh water. Do not use any cosmetic products at the sites. You can apply some calamine lotion at the rash as it will help in soothing the skin. For mild-moderate symptoms a weak steroid may be used, whilst more severe cases require a higher-potency steroid.
Symptoms can be exacerbated by dryness of the skin. Use good quality moisturizers to prevent moisture loss from the skin. During the day, hydrate and protect hands with a protective and moisturizing cream. At night, repair them with creams formulated from glycolic acid—a must for improving skin texture and tone. If still the symptoms do not improve then please get a clinical examination done by a dermatologist. Fungal infections and eczema have to be ruled out then. I sincerely hope that helps. Take care and please do keep me posted on how you are doing.