I would not give tea with milk. Dogs are lactose intolerant and tea has stimulants in it that aren't good for dogs, caffeine, etc.
I would watch him and if he coughs again, I'd take him in. He may have picked up a virus.
"Having heard all of this, you may choose to look the other way.....
but you can never say again that you did not know." William Wilberforce 1759-1833