Hello edadam
You're correct about the oil getting into the combustion chamber. The instruction book (if you have it) recommends to tilt the lawnmower one way only, usually in the direction the spark plug connection is pointing (away from you). Try this, remove air filter from top of carburettor and try starting again, this will allow more air in , may help, may not. Failing that remove the sparking plug and clean with a rag soaked in petrol. Refit and try starting again. There wont be any real longterm damage, but when it does start there will be a lot of blue smoke as the oil burns off.