...if you can borrow one somewhere or unbeknownst to me have one stashed away, by all means a Roger Smith Series 2 -- solid dial version, yellow gold. Or a Daniels Millenium watch.
Failing that, I'm thinking dressy but not flashy or too complex. So, Jules Audemars Extra Thin, yes; Jules Audemars Equation of Time, no. Voutialainen Observatoire, yes, Voutilainen Decimal Repeater 10 with the open dial, no. Much as I love him/them, nothing by Vianney for the Queen. Anything by Dufour OK -- even a Grande Sonnerie (he gets a special pass on the complexity bit because the watches are so subtle and he is, after all, Dufour). A nice solid dial VC repeater would be OK -- you will be the only one that knows you have a super watch on, which will make you feel good but isn't at all pushy. Classic vintage Omega, yes; FPJ Dead Seconds Tourb, no (sorry...).
To me colored gold is just fine (of course, that's pretty much what I wear). I visualize Her Majesty as a YG person, but who knows -- lots of other metals in all of those tiaras and crowns, but I'm assuming she'll be dressed a bit more modestly for this meeting.
As mentioned elsewhere in the thread (and reflected in my first sentence above) something from the appropriate country of origin always makes a good impression, I suppose. Of course, you could then show pride in local production by going with Eva Leube -- violates my sobriety advice a bit, but is legitimately local!
If I were meeting President Obama, I might or might not try to score a RGM for the occasion. My sense is that it is more important for him to "buy American" than it is for me. I might actually wear the JLC MUT that my wife bought for me specifically because it carries something of her with it -- dressing for the memory of the occasion and to pay tribute to her belief in me rather than for the royal/presidential interaction itself.
For Putin or other despots, definitely Patek. In China, probably VC. Hugo Chavez, don't know...
Fun to think about!
Best,
Gary G