I am a UK citizen, living in the UK, married to a Japanese citizen we are currently going through the Visa application process so she can move to UK. She will however give birth (in Japan) before this process is finalised, will I be entitled to take paternity leave (paid)?
Generally speaking where your child is born doesn't seem to be a concern for the eligibility for Paternity Leave.
You can also try to fill out the form SC3 for your employer to see if the place or country of birth is actually relevant but it doesn't seem to be.
But you can check further with your employer's HR department to verify the information and other conditions that may have been imposed if any.