Thanks for the replies everyone, and for the info, IC.
DS's separation anxiety has been getting steadily better for the last 2 years, and we've been keen not to push him beyond what he's comfortable with, while encouraging him to be more independent as he's ready for it.
When I say DS wouldn't cope, I mean he gets really distressed if another person gets between him and me/DH, or if he is momentarily left in a room with someone other than me/DH/his grandmothers. It's not a temporary distress, but we see the effects persist for quite a while - sometimes for several days.
The people he gets on with are those who take the time to really listen to him, and preferably repeat what he says back to him. Neither grandfather is much good at that, which is one reason he's not keen on them. He has particular issues with my dad because he's constantly nagging DS to give him a smile / come and show him things / give him a kiss etc, which DS hates.
DS is incredibly chatty, but unfortunately his diction is pretty unclear (he's been referred to speech therapy) and I can't remember a time when any adult other than us or grandparents has actually been able to understand what he has said. He can get quite frustrated when people can't understand him.
DS also has a strong sense of personal space and hates it if anyone other than us/grandmothers tries to touch him or come too close to him. He is reluctant to kiss even his grandmothers and has told us he would rather wave hello/goodbye to them (though he is happy to stand close to them once he's warmed up to them). I have to say I have a lot of sympathy there - I feel like that about most people myself! :)
He is okay with other children as long as they (a) don't touch the toys he's playing with, (b) don't come too close to him and (c) don't get in between him and me/DH. Obviously staying for a while with anyone else who has children would involve all of those issues.
We spend lots of time with other children, and see the same few adults and children at least twice a week, and meet up with them to play at other points. He will talk about them, but never to them and usually ignores the adults if they try to chat to him (which they rarely do).
When he has been in situations where he's been 'forced' to have closer contact with people, it's had really negative effects. We spent a day with friends who had a friendly 3 year old who kept trying to give DS toys to play with. After the child had approached DS a couple of times, DS then screamed and begged to sit on my lap & have milk every time the child approached within 5 feet of him. Afterwards his general separation anxiety (with us as well as anxiety about other people) regressed by about 5 months, and it took about another 3 months to get back to the position we had been in before. I really don't want to go through all that again.
I think if we had a child-free friend we could see at least 3 times a week for a month or two he'd be fine, but no-one we know has the time to do that even once a week, let alone more often. Your au pair solution would probably work, IC, but I'm not sure I could cope with someone else staying in my house - even the possibility of grandparents having to stay for a few days makes my heart sink! 
DS would hate nursery (we're not planning to send him, and wouldn't have planned to even if he didn't have separation anxiety issues). I think he'll be fine with a sibling though - apart from the separation anxiety he's a very easy-going child who doesn't need a lot of attention or 'entertaining'.
My mum is retiring soon, so I should be able to get them to visit more often in the run-up to the birth. It's that period between labour starting and them getting here that's the tricky bit.
I think the doula idea is definitely worth investigating. I had an unexpected overnight hospital admission at the weekend, and DS was completely unfazed by it, so I know he'd be fine as long as DH is around. I'm hoping for a fast delivery - DS arrived only 5 hours after contractions started, but I know it may not be as easy next time!