If the instruments (call them $Z$) for $X$ are not cointegrated with $Y$ but are integrated, then running a regression of $Y$ on $Z$ will yield spurious results. The regressand $Y$ will be diverging from any linear combination of the regressors $Z$, by the definition of absence of cointegration.
Regressing $\Delta Y$ on $\Delta Z$ is another matter as these are stationary (supposing that the order of integration of $Y$ and $Z$ is both 1). However, if $Y$ adjusts towards the equilibrium with $X$ (i.e. there is an error correction term in the equation for $Y$ as a function of $X$), this will not be captured in the regression of $\Delta Y$ on $\Delta Z$, and so there will be an omitted variable bias.
But before modelling $Y$ and $Z$, I would go back and question the choice of the instruments $Z$. The lack of cointegration between $Y$ and $Z$ implies also that $X$ and $Z$ are not cointegrated, which means $X$ will be diverging from any linear combination of $Z$. Hence, the instruments are poorly suited. This also suggests that what you describe by
yet I have access to data which has high correlation to the endogeneous regressor, X
may in fact be spurious correlation.