No, there’s no difference in how long a zigbee bulb takes to act as a repeater vs any other device class. It should happen right away.
That said, there are two separate issues.
First, most Zigbee bulbs, including hue, just don’t make very good repeaters on a smartthings Zigbee network. They run into what is essentially a buffer overflow type condition where they will drop messages they are supposed to be repeating, causing those messages to be lost, or have to be sent again.
Ikea brand bulbs are an exception— for whatever reason, they seem to work fine as repeaters.
This Issue has been confirmed by smartthings engineering staff, and is covered in the following FAQ:
FAQ: Are Smart Bulbs Repeaters? (Updated 2019: the new answer is yes, but may be inconsistent)
Second, to get the mailbox sensor to use one of your other repeaters, it has to be added to your network last. Otherwise, it may not realize the repeaters exist. The alternative to this is to force a Zigbee heal. Leave all of your Zigbee devices, including the sensor, on power except the hub. Unplug the hub, and if it has batteries, take those out. Leave the hub off power for about 20 minutes. this will cause the other devices to go into “panic mode“ because they cannot reach the hub. Then when the hub is back on power, all the other devices will check their neighbors and choose the best routing. That’s how you get them to start using new repeaters.
That process may be the one you were thinking about: it can take a while for all the tables to get rebuilt, so you may not see the results until the next day. But it’s nothing specific to bulbs: it would apply to any Zigbee Device.