IThe cost of the overhaul depends on your shop; just how much work they do for a "complete" tune up, and their rates for it. With something like a Sports, there's not too much that needs going over. Most of the work is in the Sturmey AW hub; next biggest PITA aside from that is replacing brake cables, as the earlier ones had calipers that use the annoying double-ended set-up. Good brakes, though.

If you buy one in decent shape, it shouldn't need new brakes. New pads might be a good idea, but in either case, you wouldn't need new wheels as a result of changing brakes. The trick is, you'll want to find one in half-decent shape. Depending on your market, you can still find nice ones for cheap on CL. In my area, the prices on any old Brit bike have skyrocketed, but the step-thru bikes are still quite affordable.
By you, I guess:
Raleigh Sports 3 Speed girls bicycle

The thing about these bikes is, a lot of the parts aren't compatible with other parts. So, if you want to change the handlebars, you gotta change the stem. The headset, BB, and cranks are basically not compatible with anything else, unless you replace the fork or go to extreme "MacGyver" style measures with the crankset. Front hub is undersized, tires are the 590 size, which is common worldwide but many US shops don't even carry them...and with very few exceptions, they are all 590x37mm. The good news is, they are very well-thought out bikes, and the only thing i might change would be the rear sprocket; they're geared kind of high for hilly areas. (But they're perfect for the flatlands, where I live...) But, if you look ant wahoonc, who has a '72 sports with something like 35000 miles on it, you'll realize that these bikes are built to last. Few modern bikes are the same way, at least not at the $40 pricepoint...

HTH
-Rob