If the train is running to London the market will want to focus on the most profitable market: London. Security will need to be done before boarding or a Lille shuffle, so stops will be kept to a minimum. London to Paris is 2hr, the wikipedia article quotes Paris-Milan as 4hr once the tunnel is complete, so you're looking at 6hr+ journey time (plus 1hr checkin).
A quick google shows flights with 4 operators at 2hr London (any)-Milan. Add 30-60 min transfer (depends on which part of London you start at) to Airport, 1hr check-in etc, 2hr flight, 30m to collect bags etc, 30m transfer into city and you've still got two hours to enjoy your first espresso before the train would have arrived.
I can book an easyjet this Friday 7am from Gatwick to Milan from €250. A Eurostar to Paris that Friday ranges from £194-240 and all the morning trains are booked out (apart from the 11.31 in Premier but that's >£300). It would only be competitive if there was a massive carbon tax on Air travel (very unlikely), a massive subsidy to train travel (extremely unlikely), or a massive change in the operating conditions that reduces costs (not a chance in hell). So the only people getting the train would be dedicated pro-train or anti-air people, and are there 1000 of them who all want to leave Milan at the same time for London? I doubt it.