I like very much the Maps.me and I have recomanded it to several people.
I would like to know if it's possible to search in London by the postcode or how could I add myself the postcode database (into to my phone).
http://wiki.openstreetmap.org/wiki/WikiProject_UK_Postcodes
Maps.me now lets you edit OpenStreetMap directly, so you should be able to add your own postcode directly. I don't know what kind of UI they've added, this can get kind of deep, but nothing that someone reading HN can't figure out.
Postcode boundaries, however, are still not available openly.