Seems like the plugin took a while over Cloudflare’s default 100 seconds before the request timed out, either due to some longer PHP processing time needed at the origin host/server in installation process or the update of the plugin(s), but the PHP process continues and should continue executing even though you see it.
You could temporary enable “Pause Cloudflare for this site” or switch from proxied 
to 
(DNS-only) the A www and A dermaplanepro.com (or CNAME if using CNAME approach).
Wait for few minutes.
Repeat the process of plugin updates.
Upon successfull updates, switch back to proxied mode and set to .
Otherwise, try debugging at the origin host why the request takes so long - or if you ran multiple plugin updates at the same time, rather one by one?