A fast multilevel algorithm with reduced memory requirements for the evaluation of transient near-field to far-field transforms is presented. The computational scheme is based on a hierarchical decomposition of an arbitrary shaped enclosing surface over which the near-fields of an antenna or a scatterer are given. For surface subdomains at the highest decomposition level, the angular-temporal far-field patterns are calculated directly from the known near fields over a sparse angular grid of directions and a short temporal duration. The multilevel computation comprises angular and temporal interpolations thus increasing angular resolution and temporal duration of radiation patterns while aggregating the subdomain contributions between successive decomposition levels. These steps are repeated until obtaining the transient far-field response of the whole enclosing surface. The computational complexity of the proposed algorithm is substantially lower than that of the direct evaluation. Additional reduction in memory requirements is obtained by formulating the algorithm as a marching-on-in-time windowed scheme.