Okay well copper is something that can get rid of ich and velvet which from what I've read both are similarly common these days, but if you're in the UK that's different, practices are supposedly better there. But there's still a possibility that they could have flukes or internal parasites.
Now I am a far cry from Humblefish, so make sure you do your own research on anything I'm saying here because I could very well be wrong so that's my big warning. However I do have a plan. I have my fish in QT, it's a 29 gallon. It is admittedly overstocked because I had a cross contamination screw up with my first four fish so two batches of QT are in there together. The only reason it's not giving me much of an issue I think is because in this tank I have 2 clownfish, 1 neon goby, 1 kaudern's cardinal, 1 midas blenny, and 4 clown gobies. I'm keeping a close eye on my ammonia badge and everything but I would in no way recommend that number of fish in one QT. I only ended up there from a rookie mistake.
So my plan is to run a 30 day copper treatment. My fish have all been in this tank several weeks now, I've been observing, but most importantly they're all eating. I just got my Hanna test kit for copper today, personally I wouldn't risk the chances of error with the color matching test kits, they can be fairly inaccurate.
After I run 30 days of copper, or before, not so sure yet, I'm going to run PraziPro according to the directions. This is for flukes and intestinal worms and black ich.
I also am considering running Metronidazole. This covers internal parasites, brook, and uronema.
I plan to give them some time between these medications, or maybe do the 14 day in copper route then split the fish up into 2 10 gallons for the other treatments so they're not in copper as long.
So by time you finish those three you've covered ich and velvet. Flukes and intestinal worms and black ich. And internal parasites, brook, and uronema. Those three cover pretty much everything.
Frankly I'm in no rush to get these fish into the tank. Well I really really am, but I also want to make sure that nothing will come back to bite me. I might do all of that, I might not, but here's my thinking... maybe one of my fish in there has a good immune system. It might have something internal, but it's living fine with it. It's internal so you can't see. But my fear is what if down the road I get a fish and that fish due to the fish itself or a species thing, is really susceptible to something that another fish has and isn't being bothered by. I could treat that second fish if I see it's having a problem, but if I haven't treated everything else in my display with the same assumption then as soon as that 'weaker' fish goes into the display it's sick again even if the other fish weren't ill because of it. I'd much rather take the time to fully treat everything than have to disassemble my rocks to catch all my fish and go through a 76 day fallow.
HOWEVER big however here, your list of fish, tangs, wrasses, maybe some others, they are often frequently difficult to QT because they can be sensitive to treatments. Whatever you decide on, just double check. Ask in the forums about the specific treatments for specific sigh. Tangs are often called 'ich magnets' yet some are very sensitive to treatments.
Here's a link to the different treatments and how when etc you do them.
https://www.reef2reef.com/threads/treatment-options-index.247573/
I'd suggest just reading every one of the sticky threads in the disease forum.
https://www.reef2reef.com/forums/fish-disease-treatment-and-diagnosis.771/
Because your big tank won't be cycled enough to really get a good sponge of bacteria then you need to buy some in a bottle or start the cycling now. I recently used Bio-Spira and it worked very well. It's a way to have an almost instant cycle. I don't know much about ATM colony and how well it works, but that's the idea. There's a thread going right now Bottled Bacteria Myth or Fact where a user is testing out the different bottled bacterias. Bio Spira worked for me, but I know others will recommend Dr Tims.
As for the filter, if you need to go out and buy something, as cheap as that looks, if you're going to be running medications and stuff it might be better to pick up a HOB filter because some treatments require you to run carbon as a form of removal which is just something to think of. It also helps to add a bit more water movement. In the US we recommend AquaClear filters, but I don't know if they're available in the UK. Or you could always just remove with water changes.