This is taken from the conditions of the ticket to LONDON INTL(CIV):
The outward portion must not be dated more than one day in advance of the passenger's ticket between London and the Continent. The inwards portion must not be dated more than one day after the passenger's ticket between the Continent and London
So to answer the first part of the question you will be permitted to purchase a CIV ticket to London International dated the day before your departure (I've done it several times).
As to the second part of your question, some CIV tickets to London International are advances and others are flexible, so it depends what you ask for and what you get (and I guess what is available on your route). On a recent journey to Strasbourg I had an advance ticket on the way out and a "EURO HIGH SAVER" (i.e. an off-peak single) on the way back. In any case, if you are delayed you should be conveyed on the next available train. The benefit of purchasing an Advance CIV ticket to London International in advance is, like regular advances, they're cheaper but quota controlled so limited availability.
--- old post above --- --- new post below ---
I have done it because I had an early morning Eurostar, but you can do it whatever the time your Eurostar departs. In any case no one will check
This isn't my experience at all. Every time I've bought a CIV ticket I have been asked to show my booking confirmation and travel itinerary, and a couple of times I have even asked to produce my Eurostar tickets when presenting my CIV ticket on board to check it was all legit!