Guild Wars 2

If I use an on-swap sigil & a on critical sigil, will the critical sigil not have a chance to happen until the on-swap sigil cooldown wears off?

They share an ICD, so until this ICD has completed, it will not have a chance to happen.

From the Wiki:

All sigils that have a cooldown share the same cooldown timer; while the timer is active, no other sigil with a cooldown can trigger. For example, if you are using a Sigil of Fire (flame burst on critical) and a Sigil of Battle (might stacks on swap), you will have to wait after swapping weapons (Sigil of Battle activated) before the flame burst can trigger on critical.

• Sigils will enter cooldown even if you do not gain any benefit. For example, a Sigil of Purity will trigger and enter the 10s cooldown even when the wearer has no conditions.
• All things equal, the mainhand cooldown sigil will trigger before the offhand cooldown sigil.