Hi Shabskay and welcome to the forum.
Speaking as a Series II owner, I'd agree with ManyMinors' comments.
The Series II has loads of charm, character and looks rather quirky compared with later versions. BUT they're slow and can't be hurried. It's not just the smaller engine either; the overall gearing limits both cruising speed and hill-climbing ability.
My Series II has a 948cc Minor 1000 engine, but currently retains the Series II gearbox and rear axle. The bigger engine makes no difference to speeds in each gear and a comfortable cruising speed of 50-55mph is still the car's limit, as is a maximum speed of around 60mph, with ear-defenders!
Driving a standard 803cc Series II hard, to keep up with traffic, will only shorten its mechanical life.
I'm a very low-mileage driver and seldom stray more than 50 miles from home in my Series II, usually planning a route via quieter roads if possible!
As ManyMinors mentions, some parts for Series II cars are becoming difficult to obtain, which can be frustrating at the best of times. Earlier Series II cars share some body panels and trim with the even earlier Series MM. Many of these components are not available off the shelf.
If you're considering a Series MM, its performance is not much different to that of a standard Series II. With its side-valve engine, Series MM mechanical components are fairly unique to the car and often difficult to find. There are some very skilled and knowledgeable people in the club, who use this forum, and who have been known to make their own Series MM mechanical parts from scratch.
If you want a Series II, try and find one that's had a complete set of Minor 1000 mechanicals fitted (engine, gearbox and rear axle/diff). The 948cc units are easily fitted and, with a bit of fiddling, the original Series II long gear lever can be retained. 1098cc units are pretty much as easy to fit, but retaining the long gear lever isn't quite as easy. The extra speed potential of the 1098cc set-up also really requires the later, larger 8" front brakes to be part of the package at the very least.
Sorry for waffling on so much! Good luck with your search