Oh, you sound just like me when I was pregnant with my first. I like to know exactly what's happening and when, and couldn't get my head around the fact the baby was making the decisions for me!
To be honest, if you want continuity of care when it comes to midwives, you need to move! Throughout my first pregnancy, under St Ts, I rarely saw the same community midwife twice. St G's team (I was 'yellow') was a bit better but there was no 'relationship' and none of them got to know me, and my pregnancy at all. Having said that, post-natally, I had the same midwife with both of my pregnancies, so there is some continuity there.
I know exactly what you mean about it all feeling rushed. I very much felt like I was just a number throughout both my pregnancies. However, I will say that St G's very rarely kept me waiting for my appointments, which was a good thing.
No matter how frantic it all seems, the midwives and medics I had through both of my labours were absolutely brilliant.
If you do want continuity, perhaps think of hiring a private midwife or doula. They'll be with you throughout the pregnancy and during labour. They'll know your preferences about what you do/don't want to happen, and can be your eyes and ears with the hospital staff when you're too out of it to speak! It's cheaper than a private birth and might be an option. I didn't do it, and wish I had. My husband was great, but at times having someone with us who knew what's what would have been helpful.
I've heard of one GP practice in Clapham with excellent midwifery team, who are better than most at giving you continuous care, but I can't remember its name, and you'd obviously have to be within it's catchment area for it to be of any use.