Prior to asking for a refund, consider a few things.
Yes, ask for a partial refund if:
- they missed a crucial moment
- caused you stress due to their lack of punctuality
- if it had a follow on effect on other events of the day
No, don't ask for a partial refund if:
- if they give a great collection of photos for your wedding day which captures the memories in its entirety
- they offered to stay back longer beyond the original agreement
- they offered other discount or a compensation through other means (printing / frames / etc)
Generally, a good photographer would proactively mention how they felt for being late and what they can do to compensate for it.
Now, after considering the above, if you have decided to ask for a refund, mention it to them in a positive manner. If you go hard and heavy, they may go into a shell and disagree. It's best to have an honest and polite conversation and hear what they have to say. I would then follow it up with an email to make sure you have captured the face to face conversation. If the photographer doesn't align himself with your views, then do consider if the amount of refund you are asking for is worth the staining the entire wedding photo collection with that memory.
If they behave in an unprofessional manner, definitely do leave a review and let the world know.
Keep in mind, don't spoil your memories of the day because of this, it would be a real shame. :)