I'm more of a lurker than a poster, usually, but I really felt the need to comment on this thread. @mlhs4198, I'm going to go completely against the grain and say there is a grand possibility that you will be ABSOLUTELY FINE!
I live in Spain and I flew back to the UK with my first baby, alone (!) at less than 9 weeks post-partum. I know a mother who travelled with a 2-week old, because she had to. I waited until 9 weeks as I had prior commitments here but ideally, I would have liked - and would have felt completely able - to go as soon as the baby's passport came just before he was 4 weeks old.
The passport was delivered within 10 days of doing the online application from Spain in the height of summer, with a Spanish birth certificate and documents from a non-British father. I would imagine a simple application from within the UK would be even faster.
The flights were fine both ways. Because I was alone, security at the airport was a little tricky, but in both countries staff and other passengers were falling over themselves to help me. The baby is now 6 months old and I have flown with him several times - the first flights were by far the easiest as, like you have already said, he slept and ate the whole time. Now, he's throwing himself around and shouting a lot so it's more stressful!
Living in the sun, it is easy enough to keep baby in the shade most of the day. I keep him under two forms of shade while outside, e.g. buggy with hood up under a parasol, and inside in a climate-controlled environment during the hottest hours. We've taken baby away to hotels near where we live and again, I found it easier and a bit more relaxing when he was tiny because he couldn't move of his own accord! Also, if you have family around who you trust to take him away for a couple of hours, there's no reason why you can't relax and sunbathe a bit too.
The doctors here said I could take the baby swimming whenever I wished and I didn't need to wait for immunisations. With regards to the pool though, that's where I do agree with other people. Although he's a real waterbaby now, he was not a fan at all of the unheated water when he was tiny. Still, like a PP said, the baby will very quickly let you know how he/she feels.
Spain is not a third-world country. You can buy nappies, formula and everything you need here at very similar prices to those you would find at home. As far as I know, the risk of diseases while un-immunised is not significantly higher or lower than in the UK.
The disclaimer, I suppose, is to have watertight travel insurance in the event that you or the baby were to suffer from complications following the birth. I don't see any harm or shame in making the plans - if they don't work out, they don't work out.
I firmly believe that parents should do what works for them and their family. Many people were shocked that I travelled so soon and have done so frequently since, but that is just how our life has panned out. I respect that not everyone would be comfortable doing the same.
You do you, and let them do them.