Almost certainly not the case that coal and oil are prerequisites, though things might have been harder without them, progress wouldn't necessarily have halted. Different approaches would have been taken. Growing fuel crops, for example, rather than building more mines. The important point here is that improvement in wealth and life expectancy in England started well before the industrial revolution. That revolution was enabled by this growth in wealth and longevity, and thus greater investment in technology, not vice versa.
"We claim that the exogenous decline of adult mortality at the end of the seventeenth century can be one of the causes driving both the decline of interest rate and the increase in agricultural production per acre in preindustrial England. Following the intuition of the life-cycle hypothesis, we show that the increase in adult life expectancy must have implied less farmer impatience and it could have caused more investment in nitrogen stock and land fertility, and higher production per acre. We analyse this dynamic interaction using an overlapping generation model and show that the evolution of agricultural production and capital rates of return predicted by the model coincide fairly well with their empirical pattern."
https://ideas.repec.org/p/cte/whrepe/wh016301.html
"We claim that the exogenous decline of adult mortality at the end of the seventeenth century can be one of the causes driving both the decline of interest rate and the increase in agricultural production per acre in preindustrial England. Following the intuition of the life-cycle hypothesis, we show that the increase in adult life expectancy must have implied less farmer impatience and it could have caused more investment in nitrogen stock and land fertility, and higher production per acre. We analyse this dynamic interaction using an overlapping generation model and show that the evolution of agricultural production and capital rates of return predicted by the model coincide fairly well with their empirical pattern."