I’ve adjusted the L limit screw so that the mech will move to under the largest sprocket and the chain will move to that sprocket when pulling the cable but not when I shift using the STI lever. Am I missing something simple here?

Cable tension

What? Too much?

Not enough.

Too little I would have said. How much slack is in the cable when in smallest cog?

I’ve started from the beginning now and it is going to the largest sprocket. The next problem is the indexing though, if I turn the barrel adjuster so that it changes up on one it then won’t change down on another and vise-versa 😕

Clean or replace and lube the inner cable, possibly outer/s too if you haven’t already. Sounds like it’s sticking somewhere.

Obviously ignore if you have already done this.

My rule of thumb is to always sort/check/clean/replace the cable before making any adjustments to the mech.

That’s the first thing I did was replace the cable. Could it be a bent hanger?

Check out an instructional video – do Park do them?

Start in the smallest cog at the back, do one click on the changer and adjust the cable till it goes up one cog.
Then click down again to make sure it’s smooth. If needed adjust the cable 1/2 a turn at a time.

Another problem may be the cable needs replacing, a sticking cable causes all kinds of bad shifting.

Took too long typing (edit)

I found my hanger alignment tool (can’t believe I actually had one!) and the hanger was definitely out, but I bent it so that it was aligned and it is now even worse.

Start from the smallest sprocket, adjust the limit screw so that it is at the upper limit and the chain doesn’t catch on the 2nd sprocket. Adjust the barrel adjuster so that 1st click on the shifter makes the chain go to 2nd sprocket but then when changing down on the shifter the chain won’t shift down! Adjust the barrel so that it does change down and it won’t change up 👿

adjust the limit screw so that it is at the upper limit and the chain doesn’t catch on the 2nd sprocket.

This part doesn’t really make sense, the limit screws are there purely to stop the chain coming off of the big and small sprockets, they play no part in shifting/indexing the gears. Indexing is done using the barrel adjuster on the mech once the cable is secured (or at the shifter if there isn’t one), make sure they are wound right in. Just bare in mind that the cable doesn’t need to be under a lot of tension when being secured, just enough so that there is no slack.

Have you routed the inner cable through the mech clamp properly? Mis routing it can cause the symptoms you describe

All of the above
Internally-routed cables? Check that the inner cables aren’t sticking
At this time of year; outer cable casing / bottom bracket guide full of crud
Jockey wheels worn / too much play / G&T wheels mixed up
Be careful about lubricating the inner cable, some greases get too sticky

Cheers for the suggestions so far. This is a road bike btw with 105 drivetrain. This morning I’m going to degrease and relube the rear mech and clean the bottom bracket cable guide.

Well, that did the trick! The rear mech was in a pretty dreadful mess really with loads of gunk on the jockey wheels etc. Shifting like a dream now. Thanks all!