Hey Emmo...I was going to post this last night but got side tracked,since only recently 2 of my mowers failed in this way.
Im assuming the mower runs fine when it runs?Intermitent failing.....Start by changing the plug regardless of how much spark it produces,It may be failing/over heating,very common.As for your tap,as you said cant see why it would matter.Check your module,another part that can fail intermittently.Ive seen enough of these fail over the time.As for the carby it would more likely to fail straight up or run erratically.One other thing,you said "you found it"meaning you dont know its history???Is the engine worse than you think?Meaning compression?And aside the initial failure after 10 mins,Is it flooding when your trying to do a hot start?