Conversion to OSP should not be a problem for the existing SnPb process due to the superior wetting of Pb containing solder. For lead free, it will be a slightly more challenging, especially if they require a double aided process.
There are numerous lead free solder pastes with characteristics that have been fine tuned for various cases. For OSP, and double sided reflow, it sometimes is necessary to choose a lead free solder paste with slightly more activity, since the OSP finish on the second side is compromised during the first side reflow.
The exposed copper will slightly oxidize with time, so the time delay between reflow processes can effect the amount of copper oxidation, which is an issue for lead free solder paste. Cookson, and other suppliers with a broad product offering, can recommend and apporpriate lead free paste.
Given the higher price of lead free solder, one suggestion is to examine the low silver versions, namely SACX Plus. The lower silver content helps to lower the price, while the presence of some silver and the overall formulation provides excellent performance.