If the output is across the R, then you have a simple real-axis pole;
identical to an RC circuit with the output across the C. The time-
constant of the LR network is L/R; for the RC network it's R*C.
Given an ideal, zero risetime input pulse, the output risetime (if
you're talking about 10-90% risetimes) is 2.2 times the time-constant;
or 2.2*L/R. You have probably seen this before for RC networks.
Given a finite risetime input signal, and a system with its own
risetime (in response to ideal stimulus), the rule-of-thumb that I've
seen from time to time, is that the observed risetime is approximately
the root-sum-square of the input and system risetimes:
Tr(out) = sqrt( Tr(in)^2 + Tr(sys)^2 )
This is only an approximation, of course, but I think that's what
you're looking for. The actual response in a detailed analysis would
depend on the input waveform shape, not just its risetime.