I had originally posted this question some months ago, and got back recently from the trip. The only one who said anything at all about the expiration date on my passport was the checkin agent in San Francisco at departure, who said I should get my passport renewed soon since some countries required six months validity. However, none of the foreign border agents made any remark at all of the date, although I was only in three countries: UK, Italy, and Switzerland in that order (plus one car trip Italy-to-Switzerland-back-to-Italy).

So I could have played it safe and renewed a few months early but there was no need. Somehow it just seemed wrong to renew so early and lose half a year off the 10-year term! Someone remarked, not sure if it was in this thread or elsewhere, that the UK government credits your new passport for any unexpired time; if only they did the same in the US.