You can't really alter the UI, unless you're willing to make formula fields for every single field, which isn't particularly scalable. Perhaps you might consider simply showing a "time zone" widget on the sidebar, and allow users to select a new time zone that way. Just create a new VF page and show it on the sidebar of your home page. Using "Show Custom Components On All Pages" will let you at least identify what time zone the user is currently using. It can be as simple as:
<apex:page controller="{!userTz}">
<apex:form id="form">
You are currently using <b>{!User.TimeZoneSidKey}</b>.
<apex:inputField value="{!User.TimeZoneSidKey}">
<apex:actionSupport event="onselect" oncomplete="window.top.location.reload()" />
</apex:inputField>
</apex:form>
</apex:page>
public class UserTZ {
public User user { get; set; }
public UserTZ() {
user = [SELECT TimeZoneSidKey FROM User WHERE Id = :UserInfo.getUserId()];
}
public void save() {
update user;
}
}