Yes the PIC12f510 will do the job. Just don't expect to drop the program from the 12f629 into the 12f510, they are two very different microcontrollers. If the program is a high level language like C or Basic then minimal changes might be possible?
This same question was emailed direct to me and I assumed he was referring to one of my projects RGB LED mood light driver
Since the 12F510 uses the 12 bit instruction set it's short of a couple of instruction from the 14-bit instruction set, ADDLW and SUBLW come to mind along with RETURN; you can only use RETLW which means you can't return a result in the W-register.
It only has a two level deep stack
There is no PCLATH register so the table lookup code would need rewriting
I know, it's bizzare, must have been someone important who came up with that decision, otherwise it would never have seen the light of day.
Just for good measure there is the 10F range which are also 12-bit and now they have the 8 pin 12F1822 enhanced 14-bit part which joins all the other 14-bit 16F parts