my dad works in a nuclear power plant, it wouldnt be too long id say a year or two for stuff to stop working or break down
as for vehicles and electricity theres a a good chance we might go out of power but so many stores now are built in with back up generators and sun powered shit i think we would be okay on electricity,
as for gas, even if pumps stopped working for some reason you could always find a way to suck it out of another car and put it into a jerry can, overall id sad if a zombie outbreak did happen with all our modern technology i dont think it would spread out too fast,
and i dont think wed be out of water, electricity , gas, ect for quite sometime, and by that time i think we'd mostly have the outbreak under control,
also i think movies and media make the government to be a "bad apple" like they wouldn't help anyone if shit like this went down, i dont see that happening although there would be extreme anarchy because we all know if this happend people without guide and rules would go crazy some people would probably not but theres always thoes few people who can't handle stress i think it may take a bit for the government and military to help us, but in the end i think they would be patrolling streets in tanks and guns and helping all survivors, they do it on a regular basis anyways (helping survivors),
if you want more information on what would happen with nuclear plants and such without human involvement id suggest watching the show "Life after people" as it shows a break down of things without people, and honestly from what ive gotten from the shows it would take between 5-10 years before shit starts hitting the fan with electricity and nuclear stuff so im sure wed only be going tops a month or two having to deal with zombies.