Help with Ultegra Shifters

I would go back to the basics. Put the chain on the smallest ring at the back.Loosen the cable clamp.Look to see that the rear mech is over the small ring.Turn in the barrel adjuster 'til it stops then a few turns out. Pull cable tight and tighten the clamp. Try the changer. If it doesn't change out of the small ring with one click then try the barrel adjuster outward or undo clamp and pull cable tight.Once you've got it moving with one click off the small ring you can use the barrel adjuster to fine tune it up the cassette.

Sounds like you tried a few things but need someone to see it in situ.
I wonder if some part of the derailleur is out of line from a knock.

As others have said it sounds most likely that there is too much slack in the gear cable wrt to the derailleur's starting position, i.e. the first click is essentially taking up the excess slack before the second moves things along. 

Fixing just needs removing the excess slack. To do this set any inline gear cable barrel adjuster to its middle adjustment position (so that there is some scope for subsequent on-ride fine tuning) and then use the main adjuster on the rear mech to remove the excess slack so that the mech shifts up and subsequently runs smoothly on a single click.

There's a bit of a knack to getting right, but essentially reluctant to shift up => not tight enough, to shift down => too tight. It's also much much easier with a workshop stand and quite satisfying when it's got right.

Good luck.

11 speed systems do have fine margins and sometimes it's impossible due to frame geometries to get everything working smoothly - some frames are more tolerant of cross-chaining for example. Probably more of a challenge to get the front mech set up than the rear though.

But it does sound there was simply not enough tension in the cable to begin with, or perhaps it has been routed the wrong side of the pinch bolt, or the housing isn't the right length. Basic stuff that should be checked before it goes out the door of the LBS though.

I'm voting for cable stretch here, sounds perfectly normal, back to where you got it, most shops offer some sort of 'come back in six weeks for a free tune up', it's precisely for this kind of thing.

Where do you live? There might be someone here near you who would have a look at it for you. If you're in or near Harrogate I'd be happy to.

go to your local bike shop, gear set is £15 and they can check will it over and you wlll make a friend.

Almost certainly a tension issue. Should be a single click.

You need to increase the tension by screwing out the barrel adjuster (caution - be careful not to unscrew it completely from the derailleur - that way lies the risk of stripped threads and swearing)

If you're out of adjustment using the barrel adjuster then wind it all the way back in, then back it out 1 full turn. Once you've done that you need to detach the cable from the derailleur pinch bolt and tighten the cable by yanking on it (or use a third-hand tool if you must). Then its just a question of tweaking the barrel adjster on the mech until you get reliable shifting.

Small moves of the cable tension barrel adjuster are required in order to do this. 

For further info see this manual

https://si.shimano.com/pdfs/dm/DM-RD0003-09-ENG.pdf

You need the section on SIS adjustment (Page 14 onwards)

Should be one click to move. If you push more and hold, it should move 2 or 3 cogs across.
I'd be checking the basics - that the high and low settings are correct, that the cable tension is right and the cable is not worn and is moving correctly.