Interplay of phonons, intertwined density waves, and induced spin density wave in trilayer nickelates <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="normal">P</mml:mi> <mml:msub> <mml:mi mathvariant="normal"

Lattice degrees of freedom (DoF) play a central role in correlated electron systems, strongly influencing the dynamics of the underlying charge carriers and spin excitations. In nickelates, understanding the role of lattice is essential to unravel the interplay between charge, orbital, and spin degrees of freedom in giving rise to various emergent phenomena reported recently. Here, we investigate the phononic DoF in a series of trilayer nickelates, namely $\mathrm{P}{\mathrm{r}}_{4\ensuremath{-}x}\mathrm{L}{\mathrm{a}}_{x}\mathrm{N}{\mathrm{i}}_{3}{\mathrm{O}}_{10}$ (where $x$ = 0, 0.4, 1, 2, 3.6, and 4) using temperature- and polarization-dependent Raman-scattering measurements. Our in-depth analysis of the phonon evolution with temperature and doping gives interesting insights into the behavior of these materials. All these systems undergo a metal-to-metal transition (${T}_{\mathrm{MMT}}$), characterized by the development of intertwined spin- and charge density waves. These transitions manifest as pronounced anomalies in phonon self-energy parameters, i.e., peak frequency and linewidth in the vicinity of the metal-to-metal transition. Several phonon modes show dramatic change (nearly an order of magnitude for some modes) in their softening rates across the ${T}_{\mathrm{MMT}}$, highlighting the sensitivity of the lattice dynamics to spin and charge order. These findings emphasize the crucial role of lattice DoF in mediating correlated ground states in layered nickelates.