You're really not interested in South Florida diving right? So some non-Florida options.
I tried to pick the "best of" that you can get to reasonably without a layover or red-eye. For as much inbound traffic/size of airport as there is to Orlando/Disney, oddly outbound farther south often goes through Miami first. If you are going to WDW also, there's also Epcot - they have a few smaller sharks but I understand the turtle is the more aggressive thing in the tank.
https://disneyworld.disney.go.com/events-tours/epcot/epcot-divequest/
Within about 6hrs/1 connection are Grand Cayman or Providenciales (Provo), Turks & Caicos - some of the best deep wall diving in the Caribbean. There are sharks in some areas - mostly where they're doing the Lionfish culling but we never saw one all week one year. The better dives off Provo - West Caicos and French Cay reliably see sharks on just about every dive but they're 5-6' or smaller reef sharks. Or 4' Spotted Nurse Sharks. There's one at a site that follows divers around like a puppy dog.
A couple years ago my buddy had one tail them off Grand Cayman but he was looking for a free meal in an area where they frequently cull. And they're just reef sharks so have no interest in people. Plus I believe Grand Cayman just adopted a no-cull policy on recreational public dives. Not sure if the sharks know that yet though...We dove a week on the west side and never saw a shark. Lots of Rays though. Besides the famous Stingray City dive we saw flights of Eagle Rays twice off the North Wall. Unbelievably clear water - probably 2-300' or more visibility.
Nassau, Bahamas is under a 2 hr. flight. JetBlue has a 2hr. direct flight for around $275 r/t but it's pretty sharky since Stuart Coves there is one of the original shark feed operators. The other, Bahama Divers will see sharks but doesn't bait them in. And there's just a lot of sharks in the Bahamas generally but in over 25 dives I've only seen one bigger than about 5-6'. Cove's pretty much dominates on island so there's only those two choices.
Or Freeport, Grand Bahama Island. Similar diving as Nassau, quieter, with more choice of dive operators. UNEXSO there does dolphin dives - shark dives also but at a different spot. The Theo is a good, deeper wreck - it's around 300' long IIRC. One of my favorite operators there is Fred Riger at Grand Bahama Scuba - he's been there for decades. About $100 more to fly there, Silver Airlines goes direct with a connection back thru Ft. Lauderdale. 1:15 flight time to there.
Cozumel is closer to 8hrs. from Orlando since they're going to connect thru Atlanta. It's mostly deeper drift diving. Probably the least expensive on my list, Grand Cayman or Provo tie for the most expensive dive destination I've ever been to. The Bahamas was pricey 30 years ago the first time I was there. All those smaller islands import everything so you pay for it. Cayman also fixes the US$ at .80CI$ - not sure how that affects you coming from England. T/C is also affiliated with Great Britain in some official way.
American flies to Curacao in under 6 hrs. also but check the prices, that's getting farther from Orlando. I'd almost guarantee you won't see a shark there. It can be pretty affordable also since a lot of places rent per day, not per person. Lots of shore diving there, buddies/guides aren't too hard to find and it's all very easy entry, low current diving - crystal clear water most of the time. So good that you see the rope for the site marker buoy a long way off. Lots of interesting small stuff there, turtles, seahorses, all sorts of shrimp/lobsters etc. IMO as good as or better diving as Bonaire with much easier entries - and better flights generally.
What's the 5th "Big Five"? I can only think of Great White, Tiger, Sand Tiger and Hammerhead. Oceanic Mako? chances of seeing one are pretty slim to none.
