I have lost my British passport. I have dual citizenship and have a Canadian passport. I am due to go on a short trip to Europe shortly. Can I travel out off and back into the UK using my Canadian passport.
It's generally true that a citizen of a particular country has an absolute right to (re-)enter that country if they make it to the border, regardless of what documentation they do or don't have. Specifically this is true for the UK.
However, entry may not be a straightforward process. The traveler and border officials will have to find alternative evidence of the person's identity and nationality, which may be inconvenient and time consuming. These days the UK Border Force has access to databases with names, birthdates and photos which can make it easier than it was in the past. Some countries may levy a fine on their citizens for arriving without a valid passport from that country (the US used to do this, but no more - the UK does not).
Also, if not using a road border crossing, the airline or other transport provider may not allow the traveler to board without appropriate documentation - they'll usually check against "TIMATIC". That part is where your valid Canadian passport would come in handy in this situation. It would also help prove your identity at the border, even if it doesn't prove your British nationality.
Theoretically you can enter the UK on a Canadian passport. But a Canadian passport only allows you to enter for a short visit. It doesn't allow you to live or work in the UK.
The problem will come when you return to the UK, and the immigration officer says "Why are you visiting Britain?". If you say "I live here", then they will deny you entry as Canadians need permission to live in the UK. If you claim to be a British citizen they will require you to demonstrate it. There can be alternative ways of proving your citizenship, although I am unclear how certain they are, and how much delay you might experience. Having the number of your lost passport, carrying an expired British passport and/or a birth certificate might be a good idea, as would being prepared for additional delay