You definitely need two days, one for each of the parks, and I'd get a park to park as it's not much extra but lets you do the Hogwarts Express and also you may find you want to go back on the second day and do something again. The rides are bigger and brasher than at Disney, but the lines are usually shorter.
If Cabana Bay is within your budget I'd definitely go with that as it's the easiest for the parks, the transport is more regular and starts earlier. You said you're on a budget so you probably won't get Express but if you are worried about seeing everything you can always purchase it as an extra, no need to buy in advance.
Another option may be to just get Express on the second day if you feel like you need it; combined with a park-to-park you'd be guaranteed to see the main things, but you should be OK without it, especially if you're not planning on doing some of the higher thrill rides, and/or have a few hours on arrival day to ride a few of the big hitters.
If not doing Express, then a good touring plan is wise, but Universal is easier to navigate than Disney, and you should be able to take in almost every attraction with a bit of planning. An early start really helps - if you can beat everyone else to the park you can get several things done before most guests even arrive. The parks should be open until about 9pm at night in August.
Both parks are best worked by working your way around the lagoon, there's no need to criss-cross like in Disney parks. There shouldn't be any huge long lines - the longest will be for The Hulk and Forbidden Journey in IoA, and Despicable Me, Rip-Ride-Rockit and Gringotts in USO.
My big main tip is to use Single Rider lines for many things, you will save so much time and you still wait in line together, you just don't get to sit next to each other. If your son wants to do the high thrills rides while you and your mom want to sit things out, this is another great way, as all the big coasters - Dragon Challenge, The Hulk, the Mummy, Transformers etc., all have single rider lines. If in doubt ask an attendant if there is a single rider line.
The first time you do the two main Harry Potter rides, Forbidden Journey and Escape From Gringotts, take the full line as the queues are amazingly well themed and part of the attraction, but if you want to do repeat Potters take the single rider line as you will zip past everyone else and this can be quicker than an Express pass.
There are a couple of 'secret' single rider lines too that not many people know about. Doctor Doom's Fearfall is a drop tower your son may like but probably not worth the long line, however it has a secret single rider - if you go inside the arcade next door, you can go up the stairs at the back and walk on to Dr Doom any time you like with no wait, so he could ride that while you guys were taking a bathroom break even!
The Mummy is another great ride that may be a bit too intense, but has a fast single rider line. Rip Ride Rockit has a single rider but it moves very, very slowly, and isn't that great so only worth doing if you or your son really, really want to give it a try.
Finally another great thing with single rider is you can send your son in as a 'canary' and he can ride and see if it's tame enough or too dramatic for you and your mom to enjoy, then you can always go again if he thinks you'd be OK with it.
Here are my recommendations for 'must-dos'.
IoA: Spider-Man and Cat in the Hat are the only one's everyone could do for sure. Hulk and Dragon Challenge are great, but not if you don't like drops and twists. Harry Potter Forbidden Journey should be OK, even for your Mom, but you might want your son to check it out first to make sure. Jurassic Park is great, and a little wetter and steeper than Splash Mountain, but not by too much. Dudley Do Right and Popeye however are great water rides but you will get SOAKED, DRENCHED, and be as wet as if you've been out in the rain for a day, so be warned (or wear a poncho). Cat in the Hat is a fun dark ride worth doing, and I'd see Poseidon's Adventure too as that's very clever.
USO: E.T. is my number one choice for a family ride, if you've never done it before Terminator 2:3D is great, Men In Black is good fun, as is The Simpsons.
Transformers is good but probably do as a single rider to save time, and Disaster and Twister are worth doing if you've never done them before. Despicable Me is pretty neat but not essential, and Shrek is basically a 4D movie and not much more.
Diagon Alley can consume hours of time, make sure to check out creepy Knockturn Alley, which many people miss. The Hogwarts Express is incredible, I'd pay the extra for park hopping just for that ride, so it might be worth heading straight there on arrival day. Make sure to ride it both ways, but the lines are lower late in the day so don't wait more than 25 minutes for it.
Food to try:
Butterbeer, butterbeer, butterbeer! Try it in three forms, regular soda, frozen slushee (my fave) and ice-cream. But you will see long lines for it in some places, as most people don't know that you can buy it in many locations, and if you are getting a meal at a Potter restaurant you can always get a butterbeer to go with it there.
Now on the butterbeer ice cream, here's a tip most people miss:
The ice cream shop in Diagon Alley always has long, long lines, and it does have lots of lovely flavors but most of them aren't that special and similar to what you'd find in a Coldstone or gelateria. The big reason for the lines though, is the Butterbeer ice cream, which you can only get in Universal. However most people don't realise you don't need to buy it there, as the Hopping Pot, on the far right of the courtyard as you enter Diagon Alley, is a little bar that sells all three Butterbeers, including the ice cream, and rarely has a line.
As far as eating meals, there's really no contest - The Leaky Cauldron in Diagon Alley, and the Three Broomsticks in Hogsmeade. Both reasonably priced, good food, great theming, and part of the whole Potter experience.
Finally make sure you post a trip report here when you get back!