When can I send 80 byte OP_Returns? I heard bitcoin-core 0.11 allows 80 byte OP_Returns, but I know miners operate different implementations and was wondering when it would be likely that 80 byte OP_Returns will be swiftly included in blocks.
As you can see, only 10.8% of nodes have upgraded to 0.11 and 29.3% of nodes use a pre-0.10 distribution. (Source: https://getaddr.bitnodes.io/dashboard/#user-agents)

Considering that we don't know the hashing power of these nodes and don't know your definition of "swiftly included", there's no way to exactly answer your query.

However, if you assume hashing power to be equally distributed between nodes, There's about a 10% chance that any give block will be mined by someone running 0.11, so about a 47% chance that a 0.11 block will be mined at least once an hour.