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No, not specific for coughing.

There is neither a word for it nor is it common practice for bystanders to comment any coughing, only in case you are really concerned about the cougher's condition.

As you figured out, a bystander's commit is usual for sneezing. 

A current (last 10-20 years) "developement" of "Gesundheit!" from bystanders turns into "Entschuldigung." from the sneezer in business environment = where you disturb an unknown crowd, while being with friends, the "Gesundheit! - Danke." is common.

So you could derive a "Entschuldigung." as well from the cougher because he might have disturbed the bystanders.

(updated/ specified thanks to comments)

No, not specific for coughing.

There is neither a word for it nor is it common practice for bystanders to comment any coughing, only in case you are really concerned about the cougher's condition.

As you figured out, a bystander's commit is usual for sneezing. A current "developement" of "Gesundheit!" from bystanders turns into "Entschuldigung." from the sneezer.

So you could derive a "Entschuldigung." as well from the cougher because he might have disturbed the bystanders.

No, not specific for coughing.

There is neither a word for it nor is it common practice for bystanders to comment any coughing, only in case you are really concerned about the cougher's condition.

As you figured out, a bystander's commit is usual for sneezing. 

A current (last 10-20 years) "developement" of "Gesundheit!" from bystanders turns into "Entschuldigung." from the sneezer in business environment = where you disturb an unknown crowd, while being with friends, the "Gesundheit! - Danke." is common.

So you could derive a "Entschuldigung." as well from the cougher because he might have disturbed the bystanders.

(updated/ specified thanks to comments)

1
source | link

No, not specific for coughing.

There is neither a word for it nor is it common practice for bystanders to comment any coughing, only in case you are really concerned about the cougher's condition.

As you figured out, a bystander's commit is usual for sneezing. A current "developement" of "Gesundheit!" from bystanders turns into "Entschuldigung." from the sneezer.

So you could derive a "Entschuldigung." as well from the cougher because he might have disturbed the bystanders.