If you're on a budget custom is NOT the way to go. You'll pay a ton of cash for it. Look for a prefab tank. Since that 120 pounds is for a tank and all the fixin's you're in good luck. You can set up a very nice tank for 120 pounds.
All the prices are in dollars but here's what you need.
Glass or Acrylic, glass tends to be cheaper and more resistant to scratching. Simple glass tanks are also all pretty standardized which makes shopping for equipment easier. Acrylic tends to be a little tougher but far more likely to scratch. Acrylic tanks are also usually non-standard sizes and might be harder to find gear for. Not impossible but not quite the number of choices you can get for standard glass. A 10 gallon rectangular tank is $20 (~10 pounds) at my local Petsmart. Oddly enough a 20 gallon is about the same. I'm gonna echo what the others have said and unless you're limited by space get the biggest tank you can.
For a ten or twenty gallon tank a simple hang on back (HoB) filter is your best value. I'm not sure what's available in the UK but for my money the best HoBs are the Penguin or Emperor series from Marineland. Aquarium Power Filters: Marineland Penguin BIO-Wheel Power Filters
For a tank the size you're looking at I'd go with a Penguin 100 for the ten gallon tank or a Penguin 150 for the 20 gallon. They'll set you back $20 or $25 dollars. If you want to keep a fish like a betta who doesn't appreciate the current you could also look at something like a sponge filter or small internal filter. If Penguins aren't available in your area look for a filter that has both mechanical and biological filtration. Mechanical filtration is something like foam or polyester floss to catch gunk floating in the water. Biological filtration is an area where nitrifying bacteria can colonize to remove ammonia and nitrite from the water. Bio filtration media is usually something like ceramic rings, noodles, or cylinders. Aquarium Filters & Filter Media: Fluval BIOMAX Biological Filter Media
Chemical filtration is of secondary concern. In my opinion look for something like Aquarium Filters: Hagen AquaClear Powerfilters
where it'd be simple to remove the carbon bag and add more ceramic noodles. You don't need to run carbon all the time, only to remove medications or some other tramp chemical that may be introduced.
A heater will keep your fish at the proper temperature. Tropical fish need to be maintained at an appropriate temperature, usually that's warmer than your house. I'm partial to the Visi-Therm stealths myself Aquarium Heaters: Visi-Therm Stealth Heater
A 50W will keep a ten gallon tank nice and warm and a 100W will keep a 20 gallon going good. You're looking for between 3 and 5 watts per gallon with heaters in order to maintain temps even if it gets cold in winter. Don't skimp on the heater, larger sizes cost little more than small ones, and buying a little larger heater is far cheaper than buying a completely brand new one when winter comes around and your tiny heater that's fine in summer can't keep up with things in the winter. The heater will set you back about $20. Also get a good thermometer, one of old style glass and mercury ones. They're more accurate. That'll cost you about $3.
4) Light and Hood
This is going to depend a lot on what you want in your tank. If you want live plants you'll need a better lighting system than if you are going to have fake ones. With fake plants all you need is enough light to see the fish. A simple single tube fluorescent fixture will be plenty. Those run between $25 and $35. If you want real plants expect to have to get a double or even triple tube fixture in order to get the plants enough light. Those can run from $50 to $100. If you're on a budget I'd suggest fake for the time being. You'll also need a hood, a simple glass top for your tank will be about $10 to $15.
You're going to need some supplies to keep up the tank. The first is a good testing kit that will test for Ammonia, Nitrite, Nitrate, and Ph. A lot of people like the API test kit and it costs about $15 to $20 online. You're also going to need a fish net ($2 to $3), an algae scraper (I like the magnetic ones which run about $12, a simple brush is more like $5, trust me get the magnetic one), a couple of buckets that you can pick up when full of water (I like my 2.5 gallon bucket from Target, cost $2. Get at least two buckets), and a water conditioner (a lot of people like Prime, a $5 bottle will last a good long while).
That's what you really have to have to get a tank started.
Filter: $20 to $25
Light: $30 to $75
Hood: $10 to $15
Test Kit: $20
Scraper: $5 to $12
Total: $121 to $203
That's an approximation based on US prices. What that'll run you in the UK, I really can't say all that well. That will set you from a hardware side though leaving you with decoration, fish, and water the only things you're missing.
Oh, one decor tip. If you want to go with a sand or gravel bottom buy it at a home improvement store. Around here a 20 pound bag of gravel costs $20 in a fish store. I can buy the same gravel but 50 pounds of it for $5 at the hardware store. Sand is much the same. It'll save you a good bit of cash.
So back to your actual question
I would go with either the Jewel or the Tetra. The Biube... I just hate those things. Distortion will be ugly and they're a proprietary system so you're going to have to go back to them for your equipment which can get pricey. The Rena is fairly small and it doesn't cost much to move up to the Tetra and double your water volume. The Tetra is decent, I'm not wild about the internal filter but it should get the job done. I have to say I'm pleasantly surprised to see a heater included. The Jewel has the largest water volume and a good light set up for real plants (provided you can get your own plant appropriate bulbs). The downside is it'll eat up your entire budget without being able to buy your maintenance supplies or any decor. I'd say the Tetra is probably your best bet.