The transition won't be instant, but it's already started. Hybrid cars, electric cars, wind farms, solar arrays are already here and just think where any of them were just a decade ago. The transition will actually be slower in north america because our economy and infrastructure are shaped by longtime access to cheap fuels. Third world countries with much less developed power grids and fuel distribution systems will quickly embrace widely dispersed local generation like solar and wind because of the low startup costs and lack of resistance by entrenched utilities. Look at the way cell phones were taken up in developing countries - high usership everywhere now, and for most people, the first phone they've ever had access to. They leapfrogged right over landline networks because of infrastructure cost. This won't happen as quickly in developed countries because there are highly developed, mostly paid-for networks for distribution of energy and they're relatively inexpensive...for now.
By the way, the north american electric grid is going to see some major changes too. Neither it nor oil will be devastated so much as eroded, but both will be much changed by dispersed energy generation like solar and wind. It will be a revisit of sorts to the very earliest days of electricity, when local communities and farmers had windmills or small water turbines. Power was generated and distributed on a small-scale, local basis. The future electric grid will have a blend of local production and consumption, with links to a national system so as to even out some of the surges and dips, and there'll be far fewer big, centralised power generation facilities. Hydro dams will continue but those resources are for the most part already fully exploited and new ones like Site C are the exception not the norm. If I was the owner or a major stakeholder in a utility that owned and operated power transmission infrastructure, I'd take a serious look at selling the assets. They're probably worth top dollar right now, maybe a handful of years more. They'll still be worth something as the decades pass, but lots of mods to the system will need to be made, and many people will withdraw from the grid completely, so there'll be a dwindling subscriber base to pay for those changes. Next time you read a news item about a government selloff of power grid assets, this could well be what's ultimately behind it. Well, that or some politicians with an urgent need for some quick cash to balance a budget.