This seems unlikely. There were major industrial and military uses for rechargeable batteries throughout the 20th century. Things like submarines, portable electronics, stationary fallback for critical services (phone exchanges) etc., all added up to a substantial economic interest in battery technology.
The Nikel-Cadmium chemistry was known since 1899, but the materials and production process were considered too expensive to make them practical until mid 20th century. Advanced processing such as powder sintering to increase area were simply unknown at the time when the ICE vs EV competition was in play. By the 1920s, the ICE and petrol fueled cars were capable of ranges and fuel economies we can barely reach even today with Lithium batteries, a technology that became possible only in the 70s, taking advantage of substantial lateral progress in material and chemical science.
A hypothetical world of electric vehicles would have spurred battery sales and investment by an order of magnitude or so, but the actual effects in hastening productization of high density cells would have probably been marginal and below what was required to win against the ICE, disproving the hypothesis. You can see this diminishing return of research at work today where, despite the order of magnitude increase in the battery market, progress is very still sluggish, pitted against hard, physical limits.