Hello all from England!

Brought a hybrid recently with 700x35 tyres, I'd like to get a bit more speed though and was wondering if I brought 700x32 or even 28/25 (as 700x35 slicks are rare and just too big) how that would effect the rim, would it be suitable to buy smaller tire width? Many thanks everyone

Not sure what you mean by "how that would effect the rim"?

You'll likely enjoy the smaller tires since they'll have less rolling resistance and quicker acceleration.
The down side is they'll ride harsher and need the air topped off more frequently.

My rule of the thumb is to keep your tire width a mm or 2 wider than the external width of the rim.
It seems that most hybrids are about a 22 or 23mm external width, so 25's would be the smallest. It also depends on your weight and road surface conditions.
IF you have a lot of potholes etc., you'll want to go a bit fatter to give you some "squish factor" to protect the rim.

To see how narrow a tire you can mount,look for a label or sticker on your rim that says something like "622xNN". NN will be the rim width. Compare that to this chart:
Tire Dimensions | Schwalbe North America

32's shouldn't be an issue,as well as 28's,but going down to 25's could effect the smoothness of your ride. Also note,the weight of the tire and design(or lack of) the tread is more important than the tire's width in improving performance. Schwalbe makes Marathon Supremes and Vittoria makes Voyager Hypers that come in mid to wide-ish sizes but give a high level of performance.