I was watching Curator's Corner from the British Museum on youtube and I found this video very interesting. The video was about reading Old Assyrian letters and it got me interested in the idea of reading first hand accounts that are so ancient, especially those by relatively "ordinary" people (though I am aware that people who were writing at that time were not representative of the average person from their society). So I found the blog post (linked above) which has a link to one of the letters in question. Unfortunately, I couldn't find a translation there. It seems that the curator from the video had translated the tablet, or at least had access to a translation. Are these translations available to the public in some form? If so where could I find them?
Not every tablet is translated. Not every translation is easy to find. Your case is a bit of the latter and a bit of the pdf at the bottom of this link.
The British Museum page you linked actually tells you exactly where to look for a translation or more information. The Bibliographic References section tells you what publication they pulled their information from. In this case it's "Smith, CCT IV / Cuneiform Texts from Cappadocian Tablets in the British Museum Part IV."
That is actually part IV of a study on those tablets, including translation and commentary. Unfortunately , I can't find part IV publicly available unless you belong to an institution (universities etc) that has access to this database from Cambridge. You may also have some luck with an academic library. If you're just looking for similar letters in translation, that link above takes you to part I of the same series.
If you're just looking for more cuneiform in translation then that website, ETANA is a great resource, as is the Fordham Sourcebook for the ancient Near East. Those will both have letters among many other documents. Some of the ETANA texts may be easier to peruse through this blog
The difficulty with any niche academic topic is that a lot of the resources end up either behind institutional pay walls or buried on academic websites with poor SEO, if they're online at all. This can make things very difficult when someone outside that niche academy wants to look for themselves. Fortunately, many translations are starting to trickle into the public domain and more websites are gradually being upgraded, but this is slow going.
There's also an issue of how these translations are published. Many recent translations are published in short academic papers that never really see the light of day, or as part of a more general work from a scholar who reads the original language. In the latter case, the translation may not be found if searching specifically for translations.
There are also mountains of untranslated ancient documents on clay tablets, sheets of papyrus, or other materials. It was very easy for archeologists 100+ years ago to load all of these (usually unremarkable) documents into a box. Its another more complicated task to catalog and translate everything, and that task is still ongoing for century old finds, as well as more recent ones.