The cheapest cells in china's specifications are a lie. Not a lie for those cells that came out perfect. But they got to be the cheapest cells in china by not meeting the test for perfect. Essentially, they are grade b, unable to be branded. Or worse, just made by a pirate type factory. They crank out cells, then go get a spec sheet off the internet. No quality control to speak of. They have high internal resistance, meaning when they discharge, they become a heater, and the faster you discharge a shitty cell, the hotter it gets.
They will get damaged by 30 amps every stop sign, in a 10 ah size battery. Oh sure, it will work for a few months, but not for 3 years.
Lets backtrack to what you said you need. and what it takes to get it. 20 mph takes about 400 watts if you pedal hard, and its not windy. If you pedal less, and its windy, it can take 800w. Typically, you will find its running 5 or 600 watts on average to go 20 mph. Including more like 15 mph up the hills, which takes full throttle.
with a 36v kit and 20 amps controller, you have potentially 800w. Its enough for that hill, and for that windy day. A 10 ah lithium battery, even the cheap ass one, can usually handle 20 amps, so it handles your starts, and your hills ok. Cruising 20 mph will take about 12-15 amps, so the cheap ass battery gets to take it even easier most of the time, while you pull 500w to ride 20 mph, pedaling so you can feel it. Not till you get out of breath, but pedaling enough to go about 2 mph faster than if you did not pedal.
When I commuted, I used about 35 wh/mi using 48v kits. 25 wh/mi using 36v kits. A good rule of thumb was to call a 36v battery at 20 mph, one mile per ah. You can go farther,,, by riding very slow. But on that windy day, it takes more. On that cold day, it takes more. On that cold and windy day, you get screwed. So to make it to work every day on time, you need that little bit of reserve. That makes 1 ah per mile a good rule of thumb for 36v.
But if you get that 48v kit,, you will NOT ride 20 mph. You wont.. you'll ride 25 mph or faster. It will shorten your range, and force you to buy a 20 ah battery at least.. So till you can spend a thou.. DONT buy a 48v kit.
With 36v, you can afford a 15 ah battery, which will go farther, and last longer, than a stressed out 48v 10 ah.
Right now, economy is what you want. Bear in mind, next year when you have more money, its very very easy to make upgrades to 48v, 60v, even 72v with the exact same direct drive motor you get in the 36v kit.
But if you have a thou to spend, then get a 48v kit, and a somewhat better 48v 20 ah. I bought the very cheapest 48v 20 ah, and it burned my house down. So I definitely would advise getting something better than the cheapest cell in china. On my house, the cheapest cell in china cost my insurance company $250,000.