In the span of three years, the application of large language models (LLMs) has accelerated across a multitude of professional sectors. Amid this development, a new collection of studies has manifested around leveraging LLMs for segments of the mechanical engineering (ME) field. Concurrently, it has become clear that general-purpose LLMs faced hurdles when deployed in this domain, partly due to their training on discipline-agnostic data. Accordingly, there is a recent uptick of derivative ME-specific LLMs being reported. As the research community shifts towards these new LLM-centric solutions for ME-related problems, the shift compels a deeper look at the diffusion of LLMs in this emerging landscape. Consequently, this review consolidates the diversity of ME-tailored LLMs use cases and identifies the supportive technical stacks associated with these implementations. Broadly, the review demonstrates how various categories of LLMs are re-shaping concrete aspects of engineering design, manufacturing and applied mechanics. At a more specific level, it uncovered emerging LLMs' role in boosting the intelligence of digital twins, enriching bidirectional communication within the human-cyber-physical infrastructure, advancing the development of intelligent process planning in manufacturing and facilitating inverse mechanics. It further spotlights the coupling of LLMs with other generative models for promoting efficient computer-aided conceptual design, prototyping, knowledge discovery and creativity. Finally, it revealed training modalities/infrastructures necessary for developing ME-specific language models, discussed LLMs' features that are incongruent with typical engineering workflows, and concluded with prescriptive approaches to mitigate impediments to the progressive adoption of LLMs as part of advanced intelligent solutions.