To begin at the end of your question: There are no restrictions for learning to drive a forklift for asylum seekers. You can gather informations about forklift license requirement here:
****://bit.ly/1UBvQyI (Texti ist in German, please use Google Tanslate or a comparable translation service). You don't necessarily need a driving school to get such a license, there are serveral institutions like the German TÜV which offer a special training. e.g. TÜV Süd: ****://bit.ly/1UBw9tv (German language). Some big corporations do offer a special instruction for forklift drivers too.
Kind regards, Jan