I think its OK to get a bit more clarification from them on the timing. And you can phrase it so that they know you have your own concerns and details that you need to take care of - so it won't sound like you are pointlessly nagging them.
Something to the effect of, "I will start preparing my departure from the UK (or wherever you are). To arrive in Japan to meet our agreed upon start date, I will need to apply for a visa by early February, so time quite limited" or something to that effect. Or, "In the event I am unable to obtain my visa by our agreed upon start date, will that cause any issues on your side?