The timer will not record the time if your Redmine user does not have appropriate access rights.
You need to check whether your account settings allow you to write, view, and edit your project time. To do this, in the project settings you need to select the tab «Participants» and see what role is assigned to the selected user.
You can check if a role has rights if you have access to Redmine administration. In the tab «Administration» select the item «Roles», and there you can edit all its rights, including the rights of time accounting.