When using Newey-West standard errors for my t-statistics of the slope coefficients in an OLS regression, can I still use the usual critical values for two-sided tests? (1.645 for 10% significance, 1.96 for 5% and 2.58 for 1%)
Asymptotically yes, see e.g. Andrews (1991, Econometrica) , equation (2.2) and the discussion below.
Subsequent literature has however pointed to the fact that this asymptotic result often gives poor guidance in finite samples, and hence developed so called "fixed-$b$" asymptotics. See for instance the work by Kiefer and Volgelsang [2,3].
 Kiefer, Nicholas and Vogelsang, Timothy (2002). "Heteroskedasticity-Autocorrelation Robust Standard Errors Using The Bartlett Kernel Without Truncation". Econometrica 70(5), pp.2093-2095. Stable URL. Other link to paper given by OP.
 Kiefer, N. M., and T. J. Vogelsang (2002). "Heteroskedasticity-Autocorrelation Robust Testing Using Bandwidth Equal to Sample Size". Econometric Theory 18, pp. 1350-1366. https://dx.doi.org/10.1017/S026646660218604X.