I am facing this issue from the past 6 months.
I have now got 15 outdated plugins on my WordPress setup.
When ever I try to update the plugin I see this message
The update process is starting. This process may take a while on some hosts, so please be patient.
But it does not proceed after that. It just stays on that page and nothing happens. But after like 5 minutes if I click on the Plugin page it shows new plugin version.
Can someone suggest any solution for this.