Center Oceanic Controller reported an aircraft deviated for weather without ATC clearance due to ATC not having time to respond due to their workload. The reporter stated numerous aircraft flight plans were filed through known weather areas which contributed to sector workload.

Narrative

I was working the North and South Atlantic ATOP (Advanced Technology and Procedures) position combined up to a single position; something that is normal at the start of a day shift. I was briefed that we had 3 westbound tracks filed through 2 Sigmets; there were already a couple of deviations currently taking place; but I should expect more. At just prior to XA:00 local; the sector queue became out of control with CPL/CDN [clearance] requests from Santa Maria Center; CPD and AGM requests from pilots and other system messages correlating to the weather affecting the sector; and the sector was split 3 ways to make it more manageable. There was at least one aircraft; a Aircraft X; that was deviating without a clearance simply because I could not get to his request in a timely fashion. Tracks should not be planned through known Sigmets. If the tracks must go through the Sigmets every other track should have different correlating altitudes to allow the controller to safely issue deviations. For example; if NAT A and NAT C have even altitudes; NAT B should have odd altitudes. This is something that the supervisors should be aware of when planning the tracks; and enforce to surrounding centers when the tracks are active.