Rose would've batted her eyes and asked to see it, and then told him she'd see if The Doctor could fix it for him. And then calmly wandered off. I can just see her turning up in the launch room wearing her big gritted-teeth Houston-we-have-a-problem smile with watch in hand. At which point we get The Doctor's stunned reaction, possibly a little emo, and then...

Unless The Doctor could hear the watch when handling it, he probably would have ended up having the professor open it anyway - but at least he'd have been prepared. Of course, he might have invited the professor on board prior to having him open the watch, which would still leave The Master on board the TARDIS, but at least at that point it'd have been a fight for control.