Years ago we had a flight into KORD go-around on the ILS 14R. They started flying the published missed approach procedure without advising ATC. Unfortunately approaches were also being flown to 09R. The missed approach procedure for the ILS 14R turned to the west towards the aircraft on the ILS 09R.
Well, I think the aircraft did just right. Cleard for the ILS 14R and deciding to go around, whatever else should they have flown? Assuming you were a controler there, you should have issued missed approach instructions with the ILS clearance if you wanted them to fly anything else.
For the general issue, I usually brief a visual pattern towards the circling side at 1500' AGL when cleared for a visual. But thinking about it, a word to ATC about the plan while the radios still work will be on my list from now on.
Nic