Yes, the valuation estimates on doubles in the UK tends to be significantly lower than where the gavel will strike. Given the decline in the popularity of hunting (especially in Africa) they don’t get as much demand. It’s the US Buyers that drive the prices well over the European valuation.
I think that’s true, and also that depends. In general, virtually everything at any auction of note sells way above auction estimates. The estimates are a tease to drive up demand, nothing more. This is true with Holts, Bonhams, Christies, Sotheby’s, Gavin Gardner, Joh. Springer, Amoskeague, Morphys, etc. This is particularly true with the European auction houses.
Then we go to the particular genres of guns.
12-bore guns in England are a pariah, so there are great deals to be had for Americans even if you’re the “sucker” that paid top price for every one of the best guns, they are still worth more than you paid the second they land on US soil. It’s an inefficient market, but not so inefficient that dealers are willing to get involved in it. (E.g. getting something for 20% less than true value is a good deal for a collector, but an unacceptable margin for a dealer)
But what about sub-gauges? Sheesh. They are ludicrous expensive everywhere. So damned rare by the numbers too. 96% of all British guns ever made were 12-bore, about 1% 10 bore, about 1% 16 bore. So all the rest make up a mere 2%: 2-bore, 4-bore, 8-bore, 14-bore, 20-bore, 24-bore, 28, 32, .410, and 9mm rounding out the scarce options. Even low-grade, extractor, basic box locks from tier-3 makers are selling for 8x more than their identical 12 bore equivalent.
In the double rifles, something funny is going on in London. I don’t know why, and I don’t know who, but there is a group of people bidding what I consider “stupid money” for Ludwig Borovnik Austrian Ferlachs. To me, they are very nice Austrian Ferlachs just like Franzoi, Just, and all the other makers. Why these guns are selling for gigantic money, I have absolutely no idea. They wouldn’t fetch half that price in the USA?
British Double Rifles is impossible to gauge. Literally nothing of note has come through the auctions in the past few years that I can recall. The 318WR double rifle this go around hit 10000 GBP. It was rebarreled in the 1930s, and it was converted to a rimless 318 and it was heavy. Still a nice gun, but I advised a bidder on this one and told him don’t pay that much, it just isn’t worth it. I haven’t seen a nice Watson Brothers or William Evans 470NE come through the auctions in a long time so I don’t know if they are soft or strong in the market today.
Continental double rifles are going for big money with few exceptions. Joh. Springer auctioned off a stunning Heym best grade side plated gun with 470 and 375 barrels, it went for around $40k USD landed. Not a terrible price, it would cost $80k to replace, but it would take awhile to sell in the USA at $50k simply because it’s a gigantic sum of money for any gun. A smallbore best Heym 7x65r and 375HH two barrel set went for a song in London about a year ago with a member here picking it up for a pittance…as in nearly the price the scopes, mounts, and case would cost to buy new. Go figure.
The really confusing thing are the Smallbore British stalking rifles. 10 years ago I could pick up the phone and call any of ten friends in the States and buy a 275 Rigby from between the wars for “full retail” if I was willing. $3500-$3800. A nice take down version could be had for $6000-$6500. Today, at the auctions basic guns to my eye are going for $10,000 and take-downs would surely be more. That has been a rapid increase in price over a relatively short period of time. My theory is that just like Dakotas in the USA, these Rigbys are being compared to modern replacement costs. Thus, a shooter might say “what the hell, why not buy a real vintage one since it’s less expensive than a brand new one anyway”. If true, this sentiment is doubling the interest, both collectors and shooters are vying for the same guns. Pure speculation on my part.
To your point on who is buying what, it’s hard to say for sure. Anecdotes in my friend network say that the Americans are jumping on the 12 bore best guns because the deals are quite solid if you know what you’re buying and have an agent on the ground evaluating them. The drillings, vierlings, and Germanic double rifles are definitely going back to the continent, not to America. The small bore magazine rifles are definitely coming to the USA. The other double rifles and safari caliber magazine rifles? I think they are going all over the world as our little fraternity seems to have similar tastes everywhere.
Just my ramblings for the evening.
. 
