Actually, guys, the leading 0 is not part of the area code, so it's 20 for London. This also applies for mobile numbers which have no area code but must still be prefixed by 0 if dialled within the UK. The 0 is a legacy trunk prefix which has been retained to this day.
In case why you may have wondered why the 0 must be omitted when using the the international dialing format (also known as E.164 format) such as +44 20 8123 4567 (spaces added for ease of reading with 44 being the UK's country code), it's because the trunk prefix is only applicable to calls within the UK.
In my phone contact book all of my phone numbers are in the E.164 format - even local numbers. This way I can be sure that wherever I am in the world the correct number will be dialled as the + character will be automatically converted to the country's exit code and I know I've got the rest of the number correct. So, +442081234567 will get converted to 011442081234567 if I'm in Barbados. Also I use WhatsApp a lot and the use of the E.164 format is a good way of ensuring you reduce the risk of not connecting.