I did some initial testing on this but wasn't able to replicate. In my testing accepting via webmail showed the acceptance status for the appointment even before my Inbox (in Outlook) had received the invitation email. Could you provide some more details on where the invite was generated/sent from along with the Outlook version?

The invite was external. Accepted via webmail but didnt reflect in Outlook. I had to accept the meeting in outlook by clicking on the appointment in calendar and click accept.