Happily, I managed to do this and got a proof of posting receipt, with time and date. They also stamped the envelope with some sort of official LHR stamp which I hope will suffice if the recipient quibbles about this.

many thanks for all of your detailed info — I first went to the WHSmith near north security (thought that was the only one) and the cashier told me to walk further down for the post office on the left. Soon I came to a post box and my heart sank: had he misunderstood? Luckily, I was able to reread the comments on this thread and see that there was another WHSmith nearer South security which does have a post office. So disaster averted!