Is this damage that would stop me being able to leave the country? Damage has happened today, I currently reside in the UK and have just had this tear happen.
No one can be sure whether this level of damage will be acceptable. Your passport will first be checked by the airline. If they believe it is too damaged, you won't be allowed to board. Your passport will then be checked at the Japanese border. Again, if the level of damage is deemed unacceptable, you can be rejected and sent back. I believe the former scenario is much more likely than the latter.
Unfortunately, 6 days is just a little less time than you can guarantee a replacement before your trip. You aren't eligible for the same day premium service, since that's only for renewals. You would have to use the 1 week Fast Track service. If you apply immediately you probably have a decent chance of getting the passport back in time, but obviously it can't be guaranteed.