Will my Spooky2 GeneratorX and attachments (Remotes, Digitizer, PEMF, Boosts, TENS) still work correctly when connected to power outlets with no earth? 

1. I'm not there in the UK or anything, but the answer is pretty clear. The wall wart (power pack) that comes with your Spooky2 might have a ground on it, but the output to the Spooky2 itself has no ground. It's a barrel connector with just plus and minus. Most power packs isolate the primary from the secondary (mains from low voltage output)

2. Yes, the equipment does not need an actual earth ground and the power to my USA packs only have two prongs for power which is not ground. The ground would make you much safer on other things though that use the ground but hey it was a safety feature to protect you from dying in the event of a ground fault short in equipment with metallic housing and wires that could become exposed.

Short answer, is no, it will be fine.

For more details, please check the link:
https://www.spooky2.com/forums/viewtopic.php?f=65&t=9637