Yes in Lady Margaret and Oratory you have picked two of just about the hardest state schools to get into unless you are prime minister or deputy
Oratory are just challenging the government because they dared to try to enforce a rule that says that state schools cannot select according to the parent's ability to provide support to the church, specifically, by cleaning the church and the silver
. Parents spend years ensuring their sons have the best chance of meeting the selection criteria, from making sure they are baptised by 6 months to maximising the points given for church attendance, being alter boys and cleaning etc. that then earn you the right to be in the lottery for places.
And No More Sleep is right, even with a house we owned and proof that our tenants were leaving and we were returning we could not apply to state schools until we were back resident. We initially were offered places in state schools somewhere near Heathrow, but three weeks into term a space did come up at the nearest outstanding state school because someone left and the people above us on the waiting list did not want to move their children at that stage (as did not we). However that is a gamble and there was a thread on here recently by an expat who had moved close to two state schools having been told it was likely places would come up but was in a position where three children were travelling long distances in opposite directions. It is a gamble that won't always pay off. It was for that reason that we focused on independent schools for our DDs.
Harrodian used to be a school that was easy to get into but it has become more popular locally and I understand has become more selective. Quite a lot of very rich parents, everyone assumed the Beckhams would be going there. There are threads on here with supporters and detractors though detractors possibly going on outdated reputation. If they are going to entertain your applications, go and look, my Aussie friend had her two children in the Junior School and loved it.
I would also go and look at different areas. For Harrodian for example you could rent in Barnes. Villagey feel and more family orientated than Fulham. The Aussie friend moved there at the same time as another expat friend moved to Fulham and the latter ended up deeply jealous. On the whole families tend to move further out as children get older, and people do not perceive places like Richmond as far out, easy to get into town in 20 mins on overground or slightly longer on underground, plenty of nice pubs, restaurants etc. and lots of open space, river, parks etc. for family walks, bike rides etc. as nomore says TFL can be your friend on this. Most families would make house decisions based on being close to stations with fast services to town rather than distance. That area around Wandsworth used to be known as Nappy Valley but there is now a joke that nappy valley have decamped en masse to Teddington now their children are reaching school age, and indeed the local nappy valley website has now set up new sites for Richmond and Teddington .....
By the way this is all SouthWest London information but there are equally good combinations of schools and nice places to live in other parts of London, DD currently renting near Highgate/ Hampstead and the High Street is spookily similar to Richmond, same restaurants, shops and celebrity count. And of course It is the North London yummy mummy scene Helen Fielding just wrote about........