Same dudes show up in The Oxford Dictionary of Family Names in Britain and Ireland. So in that case, at least, the name is not related to the placename “Chiltern” but rather to “Children” as in “orphanage”. The “atte/ate” is clearly a form of “at”, though.
West is a fairly common surname and means different things but one example is it was bestowed on some French soldiers who remained in England after the Battle of Hastings 1066. Well that is according to our family lore.