The situation that gave rise to my concern was pretty much the reverse of that which you described. I flew Raffles LAX-(TPE)-SIN on a DL award, but could only get Y up to BKK. Having been travelling for more than 24 hours at that point, a shower and change of clothes would have done both me and my fellow pax a lot of good prior to the jump up to BKK. As it happened, I used the loo near the E concourse to change my shirt as a stop-gap measure. It was a short flight up to BKK and, save the cramped seating (made better by a mercifully low pax load), SQ Y is better than DL domestic F, so I was fine.

The problem you raised could be solved by limited arrivals access to those coming off long-haul flights or, more practically, making it a purely arrivals lounge. I haven't experienced the BA lounge at LHR, but presume it's like the AA lounge at LHR, which is relatively small and just enough amenities, like showers, to serve as a pick-me-up for arriving pax but wouldn't be someplace you'd want to linger long.