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I'm not sure I've seen i described as a parade but apparently this it its name, according to the Household Division.
Have you seen this webpage by them? http://www.householddivision.org.uk/trooping-the-colour The photo shows tthe display area. The webpage is loaded with good information.
The Trooping of the Colour takes place at Horse Guards Parade which is a large open area by the barracks but the word "parade" is the name of the ground, not just the activity which takes place on it. They could have called it Horse Guards Square in some other country.
There is a significant procession down the Mall from Buckingham Palace by the Queen to the event, and back at the end. Lots of horses, music and carriages. A right proper show.
If you get seats in the grandstand you will be stuck there, no matter the weather, and no matter how soon you decide to head back to check out.
Work your time backwards and if your train is at 1630 you need to be checking in no later than about 1530. Eurostar is currently quoting 45 minutes (up from the 30 they used to say), and I suggest at least an hour.
I don't know yet what Underground lines will be closed or partly closed by the usual weekend engineering works. That will be available closer to the time on the TfL website.
I don't expect any changes to public transport as a result of the event. If the Circle and District are running the bottom of the circle that day, St James would be the closest Tube station and convenient to Gloucester Road.
But if you can use the Tube, I'd count on 30minutes back to the flat and similar to St Pancras, so there's another hour (1430) plus the time to chec out, and the time to wait for the tubes and walk to the tube stations, especially if in the crowd.
I'd say you could do it, and it would the event that you will remember forever, but it WILL be a touch tight.
Perhaps plan to go and then play it by ear?
Expect big crowds lining the Mall and at Horse Guards Parade, and if you move at the same times they do there will be congestion, plenty.
If you go early and stake out a good spot on the Mall, and are patient, you will be rewarded with an extraordinary show, even if you sneak off after the arrival of Her Majesty at the event, when the Mall will quiet down a lot.
If you are still there for the for the 1300 flypast (after Her Majesty has returned to Buckingham Palace and appeared on the Balcony with the family) you will have quite an impressive flypast, dependent on weather as some of the old planes are getting, well, old.
Do note - BTW - that you can watch from the Mall for free, but if you want grandstand seats your application needs to be in this month, they cost £30 per, and there is a strict dress code. And you might be on TV.
Have a look at some of the stuff on YouTube and still photos on Google and see if it what you'd all like to do...