Chelsea might be more tempted to keep him with them in view of his Euro performance. If not, and presuming Chelsea don't want to let him play for another PL team I would be amazed if we were chosen over the likes of say West Brom or Fulham. In regards to alerting new interested parties, maybe not in England, but it would have done in Europe, which could also give him options in Germany, Spain or Italy to consider. I still think the OP was on the wind up, but the response on here shows how desperate we all are for the club to move upwards in terms of transfer targets.
In terms of Chelsea's & Gilmour's requirements, pre tournament they might possibly have thought a middling Championship side would be perfect for guaranteeing him playing time I suppose, but his subsequent performance might give him the belief (and Chelsea's) to set their aims much higher.