@Thomas BRETON Thanks for reaching out and apology for the inconvenience due to this issue. It looks like you are running your function under the consumption plan. This is the expected behavior if your function app is on the Consumption plan, there can be up to a 10-minute delay in processing new blobs if a function app has gone idle and the same is documented here. In case if you observe more delay then this needs to further troubleshoot if your function app is getting started again or not. I will suggest you to open a support plan to troubleshoot this issue further. If you don't have the support plan then please let me know.
The suggestion would be leveraging the event grid trigger instead of storage trigger and configure event grid with your storage events as documented in polling and latency section.
To know more about clod start you can refer to this document.