At a simple level something like this... looks more complicated than it needs by splitting it up -- could all be on one line pretty well. Only problem with the date is, as you probably know, you can't have / characters in the filename so it repalces any / in the name with a - and removes and spaces while it is at it:
@echo off
set source="D:\logs\yourlog.txt"
if not exist "%source%" echo %date% %time% No file found at %source% >> "C:\logfile.txt" & goto :eof
REM remove /'s and replace with - in the date
set thedate=%date:/=-%
REM remove spaces from the date
set thedate=%date: =%
set dest="\\otherserver\share\logs\logfile-%thedate%.txt"
xcopy /y %source %dest%
if errorlevel 1 echo %date% %time% An error occured copying %source% to %dest% - error %errorlevel% >> "C:\logfile.txt" & goto :eof
echo %date% %time% Successful copy from %source% to %dest% >> "C:\logfile.txt"
Change the path on the source and dest line and the log file locations if you want it to record any problems running it
hth
Steve