A few thoughts/suggestions...
You can string out a sale for as long as a piece of string if you want to. People probably do it all the time just for this reason.
You could be upfront with any potential buyer and state from the outset you want/need to move at a certain time. This is called telling the truth!
You may also want to check your work contract or try and find out in advance if they will really hold you to the three month thing.
You could sell your house then get fired from your job in which case you are very unlikely to have to work any notice. (Then you could write/sell a funny article about how you had to get fired just to move.) This would be quite rock and roll and impressive.
Also check your mortgage. Are you going to use the same mortgage for the new property? A lot of mortgages allow you to move and keep the same mortgage.
Some mortgages also allow you to have a "payment holiday" where you let them know you will not be paying anything for 1 or 3 months or whatever. Once they have agreed it, you simply don't pay. The interest goes on the end of the mortgage if you see what I mean so you do pay, eventually.
This is a difficult problem. I think this is one of the reasons people have partners/get married - eg one partner supports the other partner while they make a career change or whatever.
Personally, I'm trying to do this as a single parent and it's going to be very difficult: buy/sell property, find a new job and change my son's school, somehow all at the same time. Have no idea how I'm going to do it. I only know that I am definitely going to find a way somehow.
It must be possible!