If you already have the weather widget in Notification centre and it is showing London, then right-click/control-click/two-finger tap on the weather widget, and select edit "London." Enter your closest location (e.g. Newcastle) in the available search field, and you will see Newcastle upon Tyne appear as one of the choices. Choose that, and then click Done when it appears. And then finish the edit process of the weather widget now showing Newcastle upon Tyne.
The weather widget will appear light or dark based on the time of day, and this screen shot was taken at roughly 11:40 a.m. Northeast time.