We don't have a committee but the school does encourage staff wellbeing, etc.
We currently have a Happiness Hero type thing going. Names in a hat, names picked out (includes all staff who want to take part, at whatever job/level etc they are - support staff as well as teaching staff, and management). All anonymous. Do 3 acts of kindness during the term for that person. So far I have received a cut bunch of daffodils left for me and a bar of chocolate. Others have had sweets/chocolate, recipes for their particular diet they're doing, nice message/cards for the person, hot drink waiting for them on the desk after playtime, etc.
We also have an early Friday finish rota - one staff gets to leave at playtime, so about an hour early. One of the TAs or other teaching staff covers for them. This is in additional to normal PPA, and covers all teaching staff inc management. Works out at approx one early finish per term (3 a year.)
Then, we just have a general atmosphere of wellbeing being important. People get to have time off for things like their children's school pay/assembly/sports day and others will cover happily, unpaid leave is granted for things like weddings without any hassle, the head and the deputy is there to support and back you up, extra time out of class is granted if someone genuinely needs some time to do things, requests for training or visits to other schools to see something in action is encouraged and solutions found for anything going on, teachers and Tas all share good practise and work properly as a team.
Where I work really does have a good air of wellbeing and happiness right now. I have worked ins one places where that really hasn't been the case and it makes a huge difference to your life.