Will a complete failure of the forward (upstream) O2 sensor cause my 1999 V70 Non-turbo to not start?

 

I would say no. Typically when a car is cold and first started it's in open loop mode, the O2 sensor in not being used for mixture feedback. I know one of the primary goals for manufacturers is to get in closed loop mode as soon as possible to reduce emissions but there is still some time needed to heat the O2 sensor. Do you have any trouble codes you can read?