Nuclear data is at the heart of all nuclear science and technology. It represents the interaction probabilities of neutrons with matter and is used to construct distributions in particle transport Monte Carlo codes. Due to the difficulty and cost of conducting experiments, experimental reaction data is usually sparse or not present for the majority of nuclides and reactions. It is the nuclear data evaluators task to statistically mix this experimental data with nuclear reaction model codes to produce his or her best estimate of nuclear data quantities plus uncertainty. The challenging task of propagating this uncertainty to their quantities of interest falls to the nuclear engineer and scientist. The current most rigorous method for the propagation of nuclear data uncertainty in neutron transport is the so-called Total Monte Carlo method. Total Monte Carlo is effectively a 2-step Monte Carlo loop: in the outer loop, a random instance of the nuclear data is produced which is then used to construct the distributions of the inner loop. Although this provides a rigorous method for propagation, it is very inefficient for fusion due to the large number of nuclides that must be considered. In this paper, we present a method for propagation that integrates these two loops, allowing for random distributions to be constructed ad hoc within the transport code and the Total Monte Carlo mean to be calculated in a greatly reduced computational effort. We propose a method based on probability bound analysis which may be used in conjunction with this to provide an upper and lower bound of uncertainty. We believe that these methods will also be useful outside of fusion: for the wider particle transport community.