Delay function in ARM programming
Many compilers and libraries support the ARM family.  I wouldnt waste time searching for a single solution defined in a single filename.  That is most certainly not "the" solution for "ARM".  

As with AVR and most other processors, certainly microcontrollers, you have the choice of blocking in a foreground loop either using a calibrated loop, or polling a timer.  You can also use a timer based interrupt if need be.  It depends on what you are needing to perform a delay for and what else might be going on during that delay.  Depending on the specific chip and amount of time you want to delay you might be able to put the chip asleep for that period of time.  

If you are using a library for other things like SPI, I2C, etc for the target processor and toolchain then you likely have timer routines as well and can just use those.  If you are rolling your own (most portable but also the most work) then look at the timers, in the long run it will be easier to maintain code that is based on a timer rather than based on a calibrated (count to N) loop.

There are a number of toolchains and libraries tailored to the NXP LPC family in particular if that is the path you wish to take.

In no way, shape, or form.  Is there a single solution that you should limit yourself to.  Not for ARM not for AVR, not for any of them.