Hi writer my ds2 (who's now 3) had this procedure just after his first birthday. It was picked up in the hospital after my c-section (at 39 weeks) and at the 6 months check there was still only one testicle so we were referred to hospital where the consultant gave us the same possible outcomes as you've been given. He also pointed out that the procedure was vital as if there was an undescended testicle and they didn't remove it that it could likely become cancerous.
We arrived about 8am on the day of surgery and after checks I think he went in about 10. I was allowed to carry him in to the room for the GA and stay with him whilst they did it. I'm glad I was there for him but it was definitely the worst bit for me as he screamed and tried to fight it, although they did tell me that he was one of the noisiest they'd seen for a long time (he still is stubborn!), so your ds may well be different. He went under really quickly though, and the staff were great.
He was in surgery for only about an hour if that and the surgeon came out to speak to us as soon as he could. They found ds didn't have a left testicle at all, and that they think it had disappeared very early in the pregnancy, which wasn't really the outcome we were hoping for, but it meant that it had been a quick procedure as they hadn't had to bring one down. They had made a cut in his scrotum to put a stitch in to keep the right testicle in place.
We are allowed into recovery to be there when ds woke up. He was very groggy but I was allowed to feed him straight away (I was still breastfeeding but I reckon a bottle would work just as well). He did get a bit upset as I think he was a bit disoriented but he soon calmed down and had a long sleep on me. The boy (who was 15 months) in the next bed was absolutely fine after his GA and munched his way through a whole lunch box of food when he woke up so I think children seem to react differently!
Ds recovered really quickly and we were allowed to leave hospital by about 3 pm so we were not in hospital long at all. DS slept a lot that afternoon and evening but he didn't seem to be in any pain. They gave us paracetamol just in case but he only had one dose. He was walking about everywhere the next day like nothing had happened 
He just had a small dressing on his belly button and it healed incredibly quickly. You can't see any scars on him at all.
He had to go back for a check up at the hospital with the surgeon about a month after, who explained what he'd done again and even showed us the photos (bizarrely he even got me to take a photo of them with my phone as a record for us, which I'm not entirely sure why - I'd completely forgotten until I was scrolling through my pics the other day, luckily the DC weren't looking too 
). He told us that it wouldn't affect him at all and that fertility should be at the same level as if he'd had two testicles. He did say that when he got to 18 there was the option of an implant but that he wouldn't recommend it as there was a risk it could damage the remaining testicle.
We were upset for him to start with, but he is completely healthy otherwise and I honestly can't see much difference between ds1 and ds2 (his remaining testicle is quite large too, which I think helps, we were told this often happens as it compensates for the loss of the other). I think people would have to be looking very hard to notice, which the surgeon pointed out why would they?
We are in New Zealand rather than the UK but I doubt there will be much difference (especially as most of the nurses were British). It was a lot quicker and easier for him than I thought it would be and DS bounced back incredibly quickly, you wouldn't have known he'd been in hospital by the next day.
Feel free to ask any questions here or pm me.