I work for a training provider specialising in f/lift, and we recommend these steps to company owners
Ensure your drivers complete a daily log sheet for the vehicle. Regularly inspect the trucks (preferably at the end of every shift) for damage. Train your staff (HSE dictates they don't need an official license & if they're working within your premises, they only need an in-house certificate. It's deemed best practice to give them an annual refresher). If funds allow, reduce the speed of your trucks. Ultimately, get rid of anyone that doesn't adhere to your procedures, damages equipment or generally drives like a loon. These people could have you standing up in court.
We put a lot of folk through training on a weekly basis. There are a load of individuals out there with certificates/licenses looking for work.
We do very little pallet truck training (mainly due to companies approach regarding official training). If you can, try reducing the f/lift speeds similar to pallet truck speed and measure the impact on production compared to the current situation. It'll give you the answers you are looking for in your 2nd post.
Either that or sell your trucks and put everyone on a manual handling course and tell them to put their backs into it!
Good luck