Can't speak for this location, but my experience with points is typically the hotels upgrade elites if/when they have availability. My guess is individual properties have quite a bit of latitude on this. I've stayed at London Grosvenor Square eight times on points and have been upgraded eight times. All during off-season, however (and the upgrade was always to a larger executive king from a standard room).
My suggestion would be to note your request for a platinum upgrade, if available, on your reservation. Then when you check in, ask again, and, if they're reluctant to accomodate you, provide any particular good reason you might have for asking.
And by all means, understand that in some cases, upgrades are not possible. You can't expect a hotel to give away inventory when there's high occupancy, and they know they can sell the premium room you'd like at a premium price to someone else who's willing to pay.