How much to save up to study in Japan?

I am currently 23 years, living in London and have over 10k saved up at the moment. Studying Japanese in Japan is something I have thought about from time to time but was never sure was feasible. I have just started taking lessons on Italki and I plan to start classes at a language school starting from April but the one thing experts(or people who have learned languages) all seem t agree is that to learn a language, you need to be in the target country

Looking at Go Go Nihon, It appears I would be able to afford tuition but things like rent, utilities, living expenses are going to increase the cost substantially. I just wanted to see if anyone here has had experience in this area or any advice you could share before I decide if this an attainable objective for me

With due respect to Language school,I strongly suggest you to study in University with English based program, such as Ritsumeikan university
Their scholarship is generous. Even they offer full tuition reductions.

Definitely true that nothing will boost your language learning speed like studying in the country. It's also pretty much the only way to become really familiar/proficient with newly developed phrases, speech patterns, and colloquialisms.

As for how much living expenses are in Japan, I imagine it's a similar situation as in the U.K., in that it heavily depends on which part of the country you're staying in and what sort of lifestyle you want.

If, for example, you're planning to live/study in Tokyo, a budget of around 80,000 yen a month should give you a lot of studio apartment options that are within a manageable commute of downtown. 10,000 yen a month should also be enough for a commuter pass that'll cover your route to/from school. For general expenses (groceries, toiletries, utilities) 50,000 a month should be enough. As for weekend/after-school entertainment, that's going to depend heavily on what sort of things you like to do, so I can't really give any estimate.

I'm sure people will chime in saying those budget figures are too high, and you can definitely survive cheaper than that in the Tokyo area if you're willing to live farther outside downtown, have a smaller apartment, make sure there're no transfer tickets involved on your commute to work, stick to meals that stretch your grocery budget, etc. The numbers above should give you a little more flexibility in choosing where/how to live, though.

You'll also need money for move-in/initial apartment contract fees. These vary by apartment, but can be up to the equivalent of six months rent.

You'll also need money to set up the apartment. You can get a lot of kitchen supplies at the 100-yen store, including dishes and silverware, but things like furniture, bedding, and appliances (refrigerator, washing machine, microwave) are going to be more expensive. Japanese apartments often don't come with a cooking range, so you'll need to buy one, and it's not uncommon for them not to have a heater/air conditioner or even interior lighting in some rooms, so those are other things you'll need to buy. All of these prices will, of course, depend on the exact products/brands you're buying, but I wouldn't expect them to be any cheaper than such things are in the U.K.

Finally, since you asked how much you should "save up" to study in Japan, another thing to consider is whether or not you'll be working part-time while you're studying.

As a frame of reference, the starting take-home pay for most expat English teaching jobs in Japan is somewhere around 240,000 yen a month. For most people, that works out to be a feasible amount for modest, live-by-yourself in a studio apartment lifestyle, though not a particularly luxurious one.

If the sum in the above reply does not fit into your budget, I would recommend you to:
1. Look for other city. If Tokyo is not your dream, don't focus on this city if you can't afford it. Even though you don't notice a big difference in food and transportation costs, you can save a lot on living expenses.
2. Ask a school if they have a dormitory. Even though it is not that common for language schools to have a student accommodation, some of them have it. (Although the number of rooms can be much lower than the number of students).
3. In case you are not lucky to get a place in a school dormitory, try a private dormitory. Both times, I lived in Japan, I stayed at shared houses and that was much more convenient and affordable than renting an apartment.
A. The rooms, as a rule, are partly or fully furnished. The common spaces are fully furnished. In the houses I lived they allowed me to use even the dishes. I had to spend money only for the bed set (blanket etc.)
B. There are no key money, lock change fee etc. Although they still charge you for move out cleaning, deposit and sometimes for contract, but it is usually not more than a month rent.
C. The main thing you should understand before agreeing for a share house, is that here you get only a room for your own use, bath, shower and kitchen are what you have to be ready to share with other people.
4. You can always take a part time job.

@Kim
But Japanese universities are ridiculously hard to gain admission, Ritsumeikan specific is one of the most competitive universities in the country, language schools at least are easier to attend.