The time will only start in June if you can evidence it. Do you have a copy of your travel ticket to the U.K.? Bank statements showing activity in the U.K. at the end of 2012. Any rental/hotel documentation from that time?
Otherwise the likely start date will be July 13.
Your partner will need to obtain a PR Card as well in order to be able to apply for naturalisation (and as evidence of status in the U.K. more generally).
Perhaps consider the European Passport Return Service, if eligible. Keep copies of everything sent.
https://www.gov.uk/government/collectio ... rn-service
Once you have your PR documented, you can apply to register your son as a British citizen- assuming he was born in the U.K.. The Home Office will only need to see PR evidence from one parent (simpler if it is the mother)- not both.
This is not intended to be legal or professional advice in any jurisdiction.