I usually write pretty long, comprehensive posts to try to answer every question a first-timer has, but... quick pointers it is! :wink:
Is the rear swingarm of the Mongoose steel or aluminum? If you have no idea, take a magnet to it and tell us if it sticks.
There are two ways you can go about it: chain drive or hub motor. Most people use hub motors, which are direct drive motors that sit in the wheel's hub. They're very rugged and dead easy to install. The biggest drawback of a direct-drive hub motor is its weight; to have the same power output as a faster-spinning geared motor, a hub motor needs to have more torque, and to do that it bust be heavier. Of course, 10lbs isn't a whole lot in perspective when you add a 200lb rider. Chain drives generally are lighter, more efficient, and can be noisier than a hub setup (though not necessarily on all counts). The problem is that they're more difficult for the average commuter to build and maintain.
You can run a motor through the bottom bracket (BB) of the bike. Several folks have very successful builds using a BB drive, and there are some bb kits out there. You can use the chainrings as a reduction gear and then use the rear gears to shift. It might be better to use an internally-geared hub like a Nexus 7 than a traditional derailleur, though, but someone else would know better than I.
When you say "it should pack a punch", what exactly do you mean? How much power do you need? Do you want great acceleration, or a high top speed, or both? What sort of cruising speed do you want for your commute, and what sort of top speed would you want for the weekends? And what sort of range do you want (you will easily find uses for extra range!)?
I'm a hub motor guy myself. There's a wide range of them out there, from little cheap brushed ones to the 25lb monster that is the Crystalyte x5 (aka 5300 aka Phoenix). Generally the power ratings on them are very conservative, and they tolerate overvolting exceptionally well. They take a real beating.
So, a a quick primer on hub motors. The most popular are the Crystalyte hubs, which come in two lines: the 400-series and the 5300-series. The 400s are 12lbs and can deliver 1kw-1.5kw of power continuously and 2.5kw for short bursts without overheating. They'll do 30-35mph without too much trouble, and they can hit 40 for short periods. The x5s, on the other hand, are meant for high-power bikes. They'll easily do 40mph, and in some extreme cases can do 60mph. With the right batteries, they can have insane acceleration and hill climbing. Both motors are virtually silent (my freewheel ratcheting is louder than my 5303). A 400 or x5 will set you back $300 or $450 respectively.
Then there's the BMC/Puma, which is sort of in the middle. It is internally geared, making it (slightly) louder than a direct-drive hub. It's light (9lbs), and has incredible torque. Puma setups generally top out in the 30mph range,and I'm not sure how far they can be pushed. Still, in the 20-30mph range, their acceleration is only matched by really powerful x5 setups. Unfortunately, they cost a mint, at somewhere above $500.
And there are several smaller motors and kits, like brushed Crystalyte motors, Wilderness Energy, Golden Motor, and the like. They're less expensive, less powerful, but fun nonetheless.
And... your battery. The battery often takes up half the budget, and it has a huge impact on the range and speed you can achieve. Nowadays, for a new build, it's best to go straight for lithium batteries, with cheap Chinese LiFePO4 on the market. Some of it is iffy, and some is pretty nice. The Chinese supplier most folks here have had good experiences with is a fellow named Ping, who assembles lithium packs and sells them on ebay. They're amazing for the price, as long as you don't need too much power from them (like for an x5). Ping packs are often called "duct tape" batteries because he wraps them in duct tape instead of heatshrink.
My boilerplate recommendation is a Crystalyte 400-series hub motor, a 48v 10ah-20ah LiFePO4 "duct tape" battery, and a 72v controller for a commuter bike with some kick and some range. You can hit 30mph with such a setup, and depending on the battery size and your speed you could get anywhere from 20 to 40 miles of range.
If you want to do a chain drive or bottom bracket drive, someone with more expertise in that area can help you. I also suggest you take a look in the Photos and Videos thread to look at some builds and get some ideas.
I don't know much about UK suppliers, but there are several active forum members on your side of the pond who can help you out.
And of course, if you have any other questions, feel free to ask.
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