There was some use of oil for generating electric power, but most oil then (as now) was used for transportation. In the US in 1970 only 6.1% of oil was consumed for generating electricity [1]. Electricity has only recently become a viable substitute for liquid fuels in ground transportation and it still can't substitute for long distance shipping or air travel. Since electricity couldn't replace transportation fuel at the time, and there wasn't a lot of oil consumed in electricity generation in the first place, nuclear power could not do much to compensate for swiftly rising oil prices.

[1] https://digital.library.unt.edu/ark:/67531/metacrs9912/m1/1/... See table 3.