Spoke nipples have a mind of their own. If reusing nipples most will work but occasionally one will slip in the tool. I don’t think this is down to Halfords tool necessarily.
I have a stash of about 30 old nipples and probably 200 spokes saved from various bikes over the years to have spares on hand should they be needed.
Of course you can get your LBS involved, but if you take it slow fixing wheels and adjust Just a tad each time, it is a useful skill to learn. Invariably the one day you need to fix wheel at the side of the road will be a rainy bank holiday 10 miles from the nearest bike shop. So learning how to do it at home in the warm and dry will pay out later.
There are plenty of YouTube tutorials and online tips like this on Sheldon Brown‘s site. I am no master wheel builder, but as shops rarely fix while you wait I just fix it myself slowly. And if not right firstvtime round, tweak and tweak till I am happy.
You don’t need a truing wheel, although I suspect it is easier with one. I true my wheels on the frame or forks using brake gap on rim as guide to how true