How do I use retro drop down handlebars with no hoods? The preface this, I am just going to say that I am awful on a bicycle.

I have recently gotten a bike that I love, it's beautiful, and I love riding it, however the drop handlebars have no hoods so the brakes are on the tops, and I have no idea where to put my hands. I want to keep them as close to the brakes as possible because of the heavy traffic in London, and because I am not quite confident enough to be able to move my hands back in to position quickly. The problem with this is that because my hands are so close together I have less control of my bike, and I'm worried that I'm going to fall off.

Does anyone have any recommendations about what I should be doing, or am I going to need to get some new handlebars?

The Bike is a Cooper Revival 2012.
You have the option of purchasing a product such as this:


These are secured to your drop bar where your brake hoods would normally reside. They provide a hand position on the ramp of the bar without having to use drop bar brake levers. They also can install directly over your tape (but it will eventually ruin the tape, so ideally you'd install them and re-tape everything.) While it would be the cheapest and easiest option, it doesn't address the issue of your hands being away from your brake levers (which is a very important feature when riding in traffic!) 

The existing brake levers are Tektro RL720s. They're known as "interrupter" levers because they are generally installed in tandem with drop bar brake levers and they interrupt the housing, creating two sets of brake levers at two different hand positions. You could purchase any aero levers and install them on the bar along with the 720s (they need to be aero levers where the housing runs under the handlebar tape). This would require some housing and new brake cables, but it would solve both the hand position and brake access issues you've posed for ~$40.