You have basically two options in regard to moving to Canada. He can sponsor you for PR outside Canada which means you can still visit Canada while you are waiting but you can not officially move to Canada or work in Canada until you get your PR. This can take 14 months, give or take.
The other option is go to Canada as a visitor and let him sponsor you inland. It takes longer and you need to stay in Canada for the duration or you risk losing your application. However, you would be able to apply for an open work permit along with an inland application and you might have it in about 4 months. That will allow you to work of course and you should be able to access health care as well.
As for staying in the UK, as an EU citizen, you should be able to apply for a family visa for him to live in the UK with you, see http://www.visabureau.com/uk/family-permit-married.aspx However, if they vote to leave the EU as they might, I am sure they will not kick out all EU citizens in a flash. If you have been in the UK for a few years, you can probably apply for permanent residency there for you both.
