Addison Lee can provide a car just about anywhere in London on 10 minutes notice. During peak times (e.g. 1am on a Saturday night) the wait can be longer, but the cars can be booked in advance.

They're reliable, they're clean, they're insured, the drivers speak English, and the cabs all have GPS so they never get lost. They're slightly cheaper than black cabs during the day, and significantly cheaper at night.

I have a friend who was injured in an uninsured minicab. It's been a nightmare. Ever since then, it's black cabs or reputable, insured firms for me.