How can I update the brake system on this vintage Universal La Riviera ladies bike? I would be really grateful for any advise on how to go about updating the brake system on this vintage Universal La Riviera ladies bike. 

The brakes don't work properly so I can't safely ride it. The front brakes are very stiff and the back breaks have completely seized.

I tried to oil it but I think the rust damage is too extensive and it's probably too old to salvage. I'd like to install a new brake set but I'm not sure what kind to buy. Do I need the exact same model or can I change to any brake system? If I can use another brake system could you please let me know your recommendation and why it's better? 

Here are some photos:

The back brake lever has seized with rust.



Front brakes function but very poorly.



Back brakes.



Back brakes.



The bike.



A friend donated this bike to me. From what I read of the company, the bike is probably from the 1980s. It's very rusty but the frame is sound, the wheels look fine and there's no problem with the steering. I recently put on a new chain and now pedaling is smooth. There's a problem with changing gears but everything's fine if I don't try to change gears. I only use it to go to my local shops a few times a month so I'm happy as long as it's functional and safe. :)

Many thanks in advance for your time.
The brake pads are almost certainly hardened and need replacing, and the cables probably rusted and stiff, so also need replacing. Both these items are considered consumable and relatively cheap. 

Before going out and buying new ones, I would remove the cable and check the levers are free. Also check the brake calipers are opening and closing freely and spring back when released.  A spray of WD-40 or penetrating oil may be needed if really stiff, but once freed up a squirt of a light oil or chain lube will provide lubrication.  If you can get these moving then install the new, high quality pads and cables and they should work better than new.  

Replacement brake levers and calipers are easy to find if you're not after exact, era-accurate replacements.  Your local bike co-op or recycle center is likely to have brakes or a donor bike. 

Installing cables is best done with a proper cable-cutting tool. It's possible to cut cables with other tools (such as a Dremel), but it's probably better to hand it over to the LBS if you don't want to purchase a tool and cannot get access to one. 

As suggested by @Daniel, the bike is not worth a lot, and you could very quickly end up spending more on the bike than it is worth. Before spending too much time and money I suggest looking very closely at the tires, as if original these are likely perished and in need of replacement.