Hi Louise
Ancestry have the Crimean War medal roll on line. There are four - W, 2 As and an F, but no Js. One problem is that the medal roll for the Crimea is in poor condition, and so liable to transcription errors. It is true that a veteran would receive two medals - the British one inscribed with his regiment, rank and name, and an un-named Turkish medal. To use the medal rolls you need to know his regiment. There were no permanent barracks in these days, so you cannot use his location as a pointer to which regiment he joined. It would have been one just passing through.
The siege of Sebastopol only lasted 11 and a half months, not eighteen. The war lasted longer but did not always involve trench warfare. It occupied the whole of 1855, so I think you should concentrate on finding his marriage which you say took place about then.
KM