There are some answers as to why you should be cycling at whycycle. Also there are several points here , here, and here and also here amongst many others.
You can get weather reports from The Met Office including local reports for London. Or you may prefer the hour by hour reports at UK.Weather.com or the advertisement heavy accuweather
If you must travel in bad weather you have three options. (1.) Wear some breathable water resistant clothing, (2.) Keep a change of clothing at the end of your route, or (3.) take a car! If you are prepared you will not need to fret about the grey clouds and you will rarely get wet. Think about how many times you have had to run to get into shelter from the rain in a typical year. It happens but not often.
Advice for cycling in arctic conditions at bicyling for life!.
Maps at CycleMaps for minor streets or direct routes . The tfl site gives cycle-only or mixed transport routes. You can't see the cycle route instructions until you view the detailed map for your route. Multimap will give detailed street maps and also gives "directions" for motorists. You might also want to get out of the area using National Rail. I have not used it myself but cycleman may be helpful.
You can buy cheap professional looking clothing at probikekit or get excellent cheap home delivery on everything else at Wiggle
The authorative Sheldon Brown will explain the hows and whys of everything related to cycling.
You should learn how to cycle safely by reading the excellent "Cyclecraft" book by John Franklin which is available from Amazon and is introduced on the Cyclecraft site.. There is a similar book with an American bias produced by John Forester.
Everyone is required to obey the rules in the Highway Code