I think that the first thing to do is to find out what fig you have. Then you can decide how you will prune it. Some figs produce only brebas, some only main crop. There are also figs with 2 crops, but in our climate it is best to only grow one crop, at least when the trees are young (more reliable harvest, less stress for the tree, easier pruning).
If you only want to grow the main crop, then removing all the vertical branches is the way to go. For breba production, pruning half of the vertical branches would be a good strategy. There is a good video on youtube that explains how you can prune fig trees to maximize the breba crop (how to prune Desert King). The link to that video is also somewhere on this forum.
When the tree goes dormant, there is no need to pick off figs that are larger than a pea, just protecting the young branches from the cold during winter is enough. If temperatures don't drop below -5°C, I don't cover the branches.
Without a ripe fig it is difficult to tell what fig you have, but I think that White Marseilles is also quite common in the UK. Maybe your fig could be a White Marseilles?