Japanese holly. There is a problem with common names. There are at least 5 species of holly native to Japan. There are two that are quite common as bonsai, Ilex crenata and Ilex serrata. Ilex dimorphophylla is another commonly grown Japanese holly. Judging by the leaves, my guess would be Ilex crenata. See if you have a tag or label, or receipt, that came with the tree, look for the botanical name, it will start with Ilex. It will may make a difference, you should know which species you have. One resource is Wikipedia. Though Wikipedia descriptions are not even or systematic in what information they cover. For example, the information for Ilex serrata is very sparse, where Ilex crenata has much more complete information. Both seem accurate for what they do have. There are some 480 species of holly distributed world wide.
en.wikipedia.org
Birmingham, UK is inland enough that your winters probably are capable of getting pretty cold. Most of the Japanese hollies are hardy outdoors in coastal regions of UK, but I am not familiar with how cold your area gets. I'm in USA, where we get quite a bit colder. A tree in a bonsai pot is probably easiest to care for if you have a location that gets cool to cold, but stays above freezing. Ideal would be constant +1C to +4 C for at least 2 months. If you have good light warmer is okay, but you definitely want night time temperatures getting cooler than +15 C for at least 2 months of the winter.
I would wait until spring or early summer, with the pruning off of that branch. If it were mine, I would probably remove that branch flush with the trunk, but that is my taste working from you photos. Shortening the branch to the small branchlet as suggested above could also work. But as is, it is too long and straight.