Apologies in advance, this is a 'long-post warning' - read on at your own peril
I think a lot of people are looking at it as if there is one big collective pool of theme park tourists that will just suddenly surge to Universal, leaving all pre-existing parks empty behind them. By that logic, when LegoLand opened in 1996, all other parks like Alton and Chessington etc. would have ceased to exist or struggled as a result. In reality, most tourist attractions of any kind are very regional - for example, growing up we mainly went to Alton, Drayton and American Adventure because they were nearby, whereas trips to Chessington, Thorpe, Blackpool and LegoLand were much rarer because they were further away.
The same will continue to happen when Universal opens, each park will continue to serve a certain region with a percentage of guests from further-afield making a special trip. The only difference is there will another park in the market for people to choose from. Universal's goal is not to go after the very specific, regional tourism markets but to cater to the very large number of tourists (both domestic and international) that come to visit London - which is why they're building it less than an hour away in Bedford.
There are plenty of examples of Disney or Universal moving into a region and it not having an overly detrimental affect on the local theme parks. Parc Asterix is still going strong even after Disneyland Paris was built, as has Fuji-Q in Japan etc. If anything, people were ringing the death knell for almost all European parks when Disney was announced for Paris (even one's here in the UK) and yet Europa Park, Efteling, Phantasialand, Port Aventura and so many more are still going strong.
However, I can see other issues that will cause problems for the UK parks - the first being the perception of quality. I'd argue the standards of parks in the UK is quite far behind most of the rest of the world. Some of the examples I've mentioned above on mainland Europe alone massively outdo the UK's offering in terms of theming, attractions, experience, etc. I don't think any of us could expect anything on the same level of Voltron or any of the other rides at Europa Park here in the UK right now. That's due to years of under-investment and Merlin's monopoly of the theme park/tourism market.
As a result, we have a very low expectation of what theme parks are in this country, believing that a 'quality experience' is saved for big international destinations in Florida or possibly mainland Europe. This however will change when Universal shows up and demonstrates that we too here in little old Britain can have a massive, world-class tourist attraction, the likes of which we've never seen, and, over time, will raise expectations and standards.
I think smaller, family-friendly parks would actually be unaffected because they can cater to local tourists looking for an affordable day out with the kids (much like the Gulliver's parks do now). However, the larger, top parks (mainly Merlin parks) will be held to a higher standard - especially if they are charging park and accommodation prices not too dissimilar to Universal. Whilst it will always have local/regional guests to fall back on, somewhere like Thorpe Park with its somewhat minimal theming and experience (comparatively) will struggle to justify it's price and the journey to get there when you could pay a little bit extra to go Universal with its immersive theming and quality rides, food, shopping, entertainment and accommodation.
The other issue is what effect it will have on the market - will this create a boom in the tourism industry in the London area and/or whole of the UK? Will this open the floodgates to more attractions, midways or even more theme parks moving in to capitalise on the growing market? We've seen what happened when Disney moved into Orlando and what it has done to Anaheim and the wider LA/Southern California area. Suddenly it could become very cramped here, and whilst Universal might not directly compete with a lot of Merlin's parks initially, the influx of other attractions may do so.
So what can the UK parks do? I think it's important for them to establish their USP and give people a reason to come to them instead of going to Universal. Efteling I think do a great job of marketing themselves as a very quintessential, authentic local attraction leveraging the European folklore/fairytale angle and building a very immersive, well-themed experience you won't find anywhere else. I think Alton Towers are best placed to turn themselves into a 'uniquely-British' theme park unlike the massive, global entity that is Universal. I think if AT were to lean into the folklore and history of the place and of the UK at large, I think they could definitely make themselves our equivalency of Efteling. This would only help to improve the park's international draw, especially for enthusiasts who will see Alton as THE theme park to visit in the UK that you wouldn't find anywhere else. Similarly with Blackpool and it's 'very-British, old seaside funfair' vibe.
Unfortunately, I cannot seem this happening for Alton Towers or any of the Merlin parks really as that's just not what Merlin are about. In the nicest possible way, Merlin are leaches - they move into tourist hotspots, setup lots of cheap and cheerful carbon-copy midways like the dungeons or Madame Tussauds and capitalise on all the tourists already in the area. Which is why it will be interesting to see what happens when Universal moves into town as its almost the opposite of what Merlin are used to. Personally though, I don't see Merlin as the types to suddenly 'up their game' and start improving standards massively or investing heavily in bigger, more immersive attractions. They strike me more as the types to 'quite whilst they're ahead' and offload most of their 'unique assets' (such as AT, Thorpe and Chessington) and focus their efforts on their global brands (LegoLand, Madame Tussauds and the Dungeons). Either way, it's going to be very interesting to see what happens in the next 10-15 years.