viewmodel.owner is not an observable property that is why it is not updating. so make it observable.
var project = {
title: "knockout",
owner: ko.observable({
name: 'Steve',
title: 'Mr.'
})
};
var ownerTwo = {
name: 'Sarah',
title: 'Mrs.'
};
//viewmodels
var viewModel = ko.mapping.fromJS(project);
var ownerTwoViewModel = ko.mapping.fromJS(ownerTwo);
//binding
ko.applyBindings(viewModel);
//jquery event
$("#target").click(function () {
viewModel.owner(ownerTwoViewModel);
});