I don't know about the dummies bit, but getting your child into the nursery attached to the school does not mean they have any privileges when it comes to primary school applications. Also if you are returning to your previous home, then you will only be able to apply to the nearby schools (well you might be able to apply to schools that are further away, but it is unlikely you will get a place, unless they are under-subscribed). You should find a guide to primary school applications on the Tower Hamlets council web-site: www.towerhamlets.gov.uk/lgsl/1-50/13_schools_-places-_primary.aspx as well as a list of all the schools.
For nurseries, yes they all have direct applications. I don't know about school based nurseries, but private ones shouldn't mind where you live. They may want a deposit though. If you contact the Children's Information Services they should give you a list (not sure if they will e-mail though, and they might be funny about sending it to the States) www.towerhamlets.gov.uk/lgnl/education_and_learning/pre-schools1.aspx
For secondary schools, it is again geographic, I think that you can put your post-code into the Ofsted website it will tell you which ones are local to your house (you can do this for primary, and possibly nurseries too). In my experience the better the school the smaller the catchment area. You will find that most children go from their closest primary to their closest secondary, but there is nothing to stop parents from applying to several secondaries and for children to go to somewhere different to their classmates, especially if you choose a school with some selection criteria (I have no idea if there are any in Tower Hamlets though).
Re private schools they all have different application methods, but I don't think that many have a apply at birth policy. Usually there is some cut off date. Sometimes it is easier to get your child into a school out of the normal time period because when families move, there are gaps, and less competition. There seem to be six private schools, but apart from GateHouse I think they are all for Muslim children (judging from the names only)
www.schoolsnet.com/uk-schools/best-independent-private-school/top-tower-hamlets-schools/16180339/0/- 3/5/187.html
I have to say that I wouldn't expect there to be lots of good secondary schools in Tower Hamlets, given that it is a deprived area, but there could be schools slightly further afield that are possible. I wouldn't be worrying about that for a two year old though - who knows where you might live by the time you need to make a decision about that. Primary schools tend to not have quite the same issues as secondary schools (I sometimes wonder what happens when you have good primaries and then a secondary with a really bad reputation - what happens to those nice kids all of a sudden )
Try posting a thread with Tower Hamlets and Primary Schools/Nurseries in the title (at a UK friendly sort of time), and I'm sure someone with local experience will give you some ideas.
Good luck! I did the opposite move, London to the US (and then on to Canada) navigating your own country's systems is bad enough - trying to understand how it works elsewhere is really tricky.