How to update the ticket when a contact company association is changed?

If contact A was associated with a company ABC and now updated to XYZ company under the contacts page, then in this case , it will still reflect the old company's details in the contacts page. 

You could go ahead and edit the requester of the ticket to a dummy email address , save it and then change it back to the original one. By this way, even in the ticket details page, the company would reflect rightly.