Oh bother... typed a long reply and just lost it!
Basically, as far as I can tell, if you get baptised at the cathedral where I go you get the normal Sunday morning 'family' Eucharist, with the usual hymns, readings and what-have-you. The order of service is printed on a sheet, but I assume it's Common Worship, as that's what's normally used for that service (there's a later BCP service too, but the baptisms tend to be done at the CW one). I can't tell you for sure whether it's also possible to have a 'private' baptism (the website says 'For information about baptisms [etc.], contact XXX [Canon Pastor]'), but I'm not aware of any having taken place. Mind you, it's not a local parish church, so I guess it doesn't get the 'passing trade' of those who don't attend regularly but want their child baptised because it's 'the done thing'.
DD2 was baptised in May, but obviously her experience wasn't typical, as she was confirmed at the same time, so the baptism was (I was going to say 'just', but that would be reducing it to something unimportant, which it wasn't) a prelude to the confirmation in that case, and she was part of a 'job lot' (3 girls were baptised, and then about 10 children and adults were confirmed). Also she was old enough to make her own responses and we didn't have godparents... so all a bit different.
I can't say for sure whether, in normal circs, the baptisee's family would have any input into the choice of music or readings, but I assume not. Certainly the readings have always been those set for that Sunday (OT/Epistle plus Gospel) and never anything non-Biblical.
At the church we went to in the US, Baptisms were done four times a year on the feast of Christ's Baptism, Easter Vigil, Pentecost and All Saints. We were there for the last of these and members of the congregation were also invited to go to the font and renew their own baptism by taking a little water and crossing themselves. (Which created a certain amount of angst for my two girls who were not baptised and didn't know what to do...)
I realise that from the above I have made baptisms in my church sound rather joyless and 'conveyor belt' (normal service, bit of water, bish, bash, bosh and off you go...), but it's really not like that at all, and they have felt very special (not just DD's, which was obviously special, but all those I've been present at) and definitely joyful and welcoming. I guess it's just not in the nature of the particular place where I go to move very far from the set order of service (largely sung), choir, incense and candles (see above), etc., and you probably would choose to have your child baptised there only if that's the kind of thing you were after.