The DONE pin is an open drain driver which is released during the Start-Up sequence at the end of configuration. Since the pin is an open drain driver, it needs a pull-up to ensure the pin goes High. The DONE pin is expected to rise within 1 clock cycle of the Start-Up sequence. The Spartan-6 Configuration User Guide (v2.1, page 24) states that "DONE should be pulled-up with a 2.4k pull-up," however, this value is a typo in the document.The official recommendation for the pull-up on a Spartan-6 deviceis the same as for theSpartan-3 family and Virtex-4, Virtex-5, and Virtex-6 devices where a 330 ohm pull-up should be used. This recommendation allows for the DONE pin to rise within 1 clock cycle at a maximum configuration rate of 100 MHz. If a board has been populated with a 2.4k pull-up on the DONE signal, the board should work fine at slower speeds (~2MHz), but you might experience Start-Up sequence issues at increased speeds. The official recommendation is to use the 330 ohm pull-up on the DONE pin, and the 2.4k might enable to device to work just fine.
The INIT pin does not have a rise time requirement and is used to signal the start of configuration as well as a CRC, and can also be used as User-IO post configuration. The recommendation for INIT in the User Guide is to use a 4.7k pull-up. This recommendation was supplemented with a prior recommendation in this Answer Record to use a 2.4k pull-up on the INIT pin. The Xilinx Official Recommendation will still stand as using a 4.7k pull-up on INIT as the device characteristics in regards to INIT have not changed from other families.The reasoning behind the recommendation for a stronger pull-up on INIT was because INIT is a User-IO after configuration.If the INIT pin is not used in the design it will be unused, and unused pins by default have a pull-down. The voltage divider created between the pull-down and the pull-up on INIT might cause the voltage to drop lower than Vihmin at Vcco for a downstream device. For this reason, using a stronger pull-up can help ensure Vihmin is met on other devices connected to the pin. The INIT pin can also be included in the user logic and driven or tri-stated accordingly.
In short, the DONE pin pull-up recommendation of 330 ohm and INIT pin pull-up recommendation of 4.7k that Xilinx has been using for all recent FPGA families still holds true for Spartan-6 devices and these values should be used.