Speedy 300 and Fusion Pro may or may not be the right machines for you.
I would suggest when contacting vendors or looking for recommendations to provide as much as possible information about your use scenario and performance criteria. "detailed pictures into wood" and "cut out silhouettes with felt" is a bit too vague.
I've never heard of a UK company designing and producing their own laser gantry machines. The ones you mentioned resell low-end Chinese machines. The only European brand making such machines is Trotec.
All contemporary controllers use standard vector formats so using Illustrator should be perfectly fine.
That is a decent budget. You should be able to get something nice as long as you spent the money on hardware and not on seller markups.