battery is the most expensive component in any e-bike after a crank drive motor so it's not hard to see why cheaper e-bikes have a smaller battery capacity. Typically, if you ride a bike at 15mph in perfect weather condition (no headwind, no rain), the total energy required is 200W. Manufacturers estimate that the average rider can put in 100W, the bike will put in the difference, 100W. Over 1 hour, that translates into 100WH for 15 miles or 6WH per mile. I usually estimate double that, 12WH per mile to be more realistic as people may ride in adverse conditions, big hills, headwinds etc. For your commute, you'll need 36V 13AH at a minimum if you don't want to charge up every day at work.
One common theme among questions asked by newbies is how to derestrict a bike and/or add a throttle. No manufacturer will want their bikes derestricted, so the short answer is build your own bike.
Most cycle to work schemes will let you use your voucher to buy a kit.
