If you have a job offer, you should pursue a work visa. Your wife would need to be your sponsor for a spouse visa, and since she doesn't have an income, you won't meet the financial requirements. Your children should be British citizens by descent, but they will need to obtain their British passports before moving.
Get your London employer to look into a Tier 2 visa for you.
10 days is very ambitious for visa processing, but I don't know the timeframe for Tier 2
You would need to be resident in the UK on a legal visa for 5 years before you can get Indefinite Leave to Remain (ILR). Once you have that, provided you are still married to a British citizen, you will be eligible for naturalisation.