More information:

With keepalive_timeout 65, this is what /nginx_status looks like, and
new connections take 5-20 seconds to start delivering content.

Active connections: 1552
server accepts handled requests
2940 2940 38965
Reading: 0 Writing: 271 Waiting: 1281

With keepalive_timeout 0, this is what /nginx_status looks like
with the same workload, new connections start receiving
content immediately.

Active connections: 2
server accepts handled requests
5797 5797 5796
Reading: 1 Writing: 1 Waiting: 0

Can anyone suggest what to look at next? Is there some
extra logging I can turn on that might shed light on the situation?

Steve