Speed correlates directly with the ability to recognize patterns. To work on that, I'd recommend Zibbit's excellent Pattern Recognition series on Youtube. Also, something I do personally is keep a tactics database on Scid vs PC where in every puzzle I've removed all the pieces which are irrelevant to a given tactic so I can focus on memorizing patterns. It's still a work in progress but here it is in PGN format if you're interested.
Consistency and speed of tactical vision

Thanks for replying Way-of-Pain. I've checked out a couple of those videos, and they are pretty useful. The trouble is I am aware of at least the vast majority of these. When it comes to applying this knowledge, however, I still come stuck and can't see the tactic the first time. When I review the mistakes afterwards, I am able to easily solve the puzzles I went wrong on, usually on the first attempt.
I don't know, maybe it's not unusual sometimes, but I just want to come to a point where, I am able to instantly do ALL positions up to at least 2000 (rating in reference to Puzzle Rush positions) without any trouble. I guess I'll keep practising, but it'll be nice to know what people's experiences have been.

The trick apparently is to do the low rated puzzles but always solve them to the end - ideally never getting them wrong even if some take a little while

Have you tried the woodpecker method? Basically you do the same sets of tactics over and over. It works with any tactics book; you don’t have to buy the actual book (you can if you want)


hence only moving onto the next puzzle once you work each one out. eventually speed comes

After every puzzle rush, I go back through and review each problem I missed and usually the last problem I didn't have time to solve, if I ran out of time instead of getting strike 3. This lets me find my mistakes to avoid them in the future.

The trick apparently is to do the low rated puzzles but always solve them to the end - ideally never getting them wrong even if some take a little while
I'd like to try this. It sounds good. Because my "high rated" (2100) puzzles take me 10 - 25 minutes for a single puzzle. It's also challenging and fun, but I want to try your approach... I never did it... to get through like 20 puzzles (calculating them all to the end before playing move 1) in a session, but not the super easy puzzle rush ones... Something in the middle... On lichess there is a way to select "easier -300" rated puzzles, I think i'll try this.
btw, I liked this video explaining on why woodpecker method works
https://www.youtube.com/watch?v=1qicV-jaGFE
Does anyone have any advice on the best way of improving, firstly, the consistency at spotting (correct) tactical operations, and secondly, the speed at which you spot them?
I guess I'm looking more for something other than to "keep practising". I appreciate that obviously that is very much sound advice, which of course I'll keep doing, however it would be good to pick up on maybe other methods that you've guys have used to improve.
Also for additional context, I would consider myself fairly decent when it comes to more basic and intermediate tactics. I have started playing various moved in Puzzle Rush lately, and in general I can correctly solve the vast majority of positions labelled up to 1800ish at a 5 min time control (I'll have run out of time around this point). My best on that mode is a score of 34; on Survival my best is 50.
My frustration mainly is that I still get stuck on some positions that I should be able to solve easily - in fact, when I review the positions afterwards I instantly solve them. With regards to speed, I guess I would hope to improve this in conjunction with consistency.