Hi Alex,
All the other suggestions are good ones. I rebuilt this linkage on my TR4 a couple years ago. It's fitted with Weber carbs, but uses a lot of the original linkage.
Here are some other things to consider.
Start at the carbs, disconnectinig all the linkage and make sure they return well, open fully with no resistance, through the full range. Dirt and/or wear inside, or weakening springs, might cause problems.
Work backwards from there, through all the linkage, to the pedal.
I added a secondary linkage return spring at the front. This hooks to a notch I filed into the crank arm on one end and forward to one of the radiator braces on the other end. You might need to arrange differently, but the idea is to eliminate all linkage slop. I plan also to add two more "safety" springs at the carbs (to a bracket on the valve cover), which is required by most racing organizations.
Next, the end fittings on my car's linkage were heavily worn. Adjustment is possible, or it might be better to replace the ends with new. Both the ball and the fitting that snaps onto it can wear. Unfortunately, many replacements are non-adjustable and simply snap together.
If using the original, adjustable type, be sure there is a cotter pin or safety wire in the end, preventing the adjusting screw from backing out, which might let the linkage drop off. There are a series of small holes around the peripehery for the cotter pin or safety wire. If not too worn, the cotter pin or safety wire will pass through the screwdriver slot, once adjustment is correct. If heavily worn, the screwdvriver slot will be too deep for the cotter pin/safety wire to hold anything.
If you aren't concerned about originality, it's possible to make up your own linkage using 1/4" rod-ends from a hardware store, or put these types of ends onto the old linkage rods. One of my local hardware stores sells rod-ends in a variety of sizes, both male and female, both righthand and lefthand threaded. They also were a good source aluminum rod or tubing, or steel if preferred. Most racing organizations require rod-end type linkage, because it's trapped, cannot come completely apart and will at least work partially in the event of a failure.
At the firewall on your car, there are those two nylon bushings on either side, as mentioned already. Those are most likely a source of a lot of slop in the linkage and are available from the major Triumph parts sources. The clamshell cups they fit in can be hard to find, though they usually don't wear and don't need replacing.
There is also a simple brace near the linkage lever - really nothing more than a sheet metal tab that's welded to the firewall - with the shaft passing through it. The metal against metal arrangement wears too and this allows the lever shaft to move sideways, adding more slop. So, I got hold of an extra nylon bushing and welded a spare set of clamshells (which normally hold it onto the firewall at either side) onto this bracket. This added a third bushing to the shaft. It reallly helped a lot!
Finally, the roll pins in the shaft do work loose and drop out. Yet, it is necessary to separate the shaft in order to remove the pedal/linkage from the car, making welding less than desirable. So I replaced all of the pins with small socket headed screws, after enlarging th holes slightly and tapping the sleeve and shaft all the way through. A little Locktite and there ain't no way those will fall out again. This also makes disassembly a lot easier back behind the head and engine, if and when that ever becomes necessary.
Hope this helps!
Alan