A variac is simply an adjustable transformer. It allows you to set the output anywhere from 0v to the supply voltage (approx. 240v in UK). Some variacs have extra turns. Depending on how it's wired up, you can get slightly more than 240v (typically 270v) if the extra turns are used.
If you supply your variac with 110v, it will adjust over the range of 0-110v. There's no problem about using a 240v variac on a lower voltage. But be careful not to draw more than the maximum current allowed from your variac. If your variac is rated 240v 2 amps and you supply it with 110v, you can still only draw a maximum 2 amps from it at 110v, which means it can only supply about half as much power.
If you just want to slowly power up an American amplifier, you can connect your variac to 240v, set it to zero and slowly turn the dial until you reach about halfway or 110v on the scale. Just be careful not to go above halfway. No additional transformer is needed unless you need an isolating transformer. In that case, you can connect your yellow building site transformer to the output of the variac. The output from the transformer will vary from 0v at minimum to 110v when the variac is set to maximum.