Soul Warden's ability will trigger.
603.6d Normally, objects that exist immediately after an event are checked to see if the event matched any trigger conditions. [...]
That means it's the types the Signet has on the battlefield that are checked by Soul Warden's triggered ability.
604.1. Static abilities do something all the time rather than being activated or triggered. They are written as statements, and they’re simply true.
That means there isn't a moment that a non-creature artifact on the battlefield is not affected by March of the Machine.
As such, the Signet is seen as a creature by Soul Warden's triggered ability, and it will trigger.