Depends where in Paddington you're staying as it's still a pretty big area.
There are no bodybuilding gyms in the area. You might be lucky and be close to an LA fitness or Fitness First which are 2 of the better gym chains which will have loads of equipment. Worst case you will have to use one of the community gyms going under the name Better gyms. There are 5 or 4 of them in Paddington. You'll find at least one squat rack, smith machine and bench press and a few machines. Very few serious lifter or juiced up bodybuilders bodybuilders though, but there is no lunk or grunt alarm or whatever that BS is for you to worry about.