I wouldn’t bother with any other training like forklift, as you’d be very unlikely to use it once you’ve passed your hgv so it would just be wasted money.
I’d also say there’s no harm in telling your company about doing your training, as they might let you do a few shifts once you’ve passed to cover holidays and sickness, or you could be next in line to replace any drivers that leave. It’s also likely in that case that they would provide some training or send you out with someone else rather than throwing you in at the deep end, which is likely to happen if you find work elsewhere, especially if you do agency work.
Your employer can’t fire you or take any disciplinary action against you for doing extra training, if anything, they would probably be glad that they didn’t have to cover the costs
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