Researchers at the University of Pavia tested the question directly. They transplanted embryonic mouse cerebellar neurons into the developing brains of rats, a species that lives roughly twice as long as a mouse.
The transplanted mouse neurons integrated into the rat cerebellum and matured normally, keeping their mouse-specific shape and markers. Instead of dying around 18 months, the typical mouse lifespan, the neurons survived inside their rat hosts for as long as 36 months, about double the median lifespan of the donor mice.
Neuronal lifespan, at least here, tracked the lifespan of the host body rather than any internal clock set by the donor species. That suggests the limits on how long a neuron lives are set by its surroundings as much as by its own genetic program. The transplanted cells were cerebellar neurons, and whether other neuron types behave the same way across species has not been shown.

Read the original study
Magrassi et al., PNAS, 2013 · doi.org/10.1073/pnas.1217505110
A mouse neuron, living on rat time.
Source
- Magrassi, L., Leto, K., & Rossi, F. (2013). Lifespan of neurons is uncoupled from organismal lifespan. Proceedings of the National Academy of Sciences of the United States of America, 110(11), 4374–9. https://doi.org/10.1073/pnas.1217505110