Hello my name is***** but you can call me Dr. H, and I am a licensed veterinarian.
I understand that your pet was given 2 vaccinations 1 week apart. This inadvertently happens sometimes in pets and some pets have even been given double vaccines on the same day. In most cases they are completely fine, because the immune system can only respond to certain threshold and any additional vaccine won't cause an increased response.
The only thing I would consider is an increased chance for a vaccine reaction, so I would recommend monitoring (as usual) for excessive vomiting, diarrhea, and facial swelling. Vaccine reactions can occur with any vaccine and usually occur within the first 48 hrs, but can be delayed (in rare instances) for up to 7 days.
But all in all, there are no long term effects from having a pet vaccinated too early. However the vaccine that was given early can't be counted as vaccine #2, because of how the immunization process works, so technically you should wait 3-4 weeks from the original vaccine that was given at 10 weeks before the next round of shots.
Let me know if you have any questions about my recommendations, or any further concerns to discuss. Thank you.
- Dr. H