Lazy,
Not sure if you are still about or not but I was in the same position as you. I was given a cheap mountain bike (BSO). Went out on it a couple of time and it then sat in the garage.
I then bought a cheap road bike from cycle king, I think or similar. It was about £170. Matt Black Gear change on the down tube I had to change the tyres as the cheap Kendas supplied were horrendous. I almost immediately joined the local CTC and was soon riding in excess of 30 miles on that bike each week. I had the bike about a year and sold it on for £50 as I got the bug and wanted a proper road bike. You can alter bikes and keep it cheap ecpecially if you are not sure if cycling is a long term thing for you.
As regards your time Barking to Canary Wharf is extremely busy, heavy traffic, queues, plenty of lights and lots of people. You would probably cover double the distance in the same time along nice quiet country roads so do not be disheartened. This would give you a circa 12 mph average and would be just enough to keep up with our particular CTC's slow group to elevenses. Some of these people have been riding a few years. So already you have commuted to work and would be able to join your first bike club. Not bad for a days riding !