varunkapahi... he didn't use a Sovtek Big Muff. Mike Matthews hadn't purchased Sovtek and used it to make pedals until the early 1990s, several years after Cliff's death.
Cliff had an old U.S.-made Big Muff. Which version, I'm not entirely sure (the look and circuit changed a bit from year to year before EHX shut down in the 1980s, not to mention, quality control wasn't superb back then). But if it's any consolation... many bassists find it's easier to get that particular Muff sound with the green Sovtek Big Muffs. The BBM is based on it, so it should do the trick. Besides, do you really want to try hunting down an ancient U.S.-made Big Muff that will probably not sound right?
As for the distorted sound in "Anesthesia" and other songs, it's a combination of an old chrome Morley Power Wah Boost (although some say it might be a Power Wah Fuzz) and a Big Muff. Both are on, and the wah is pushed down halfway. He moves it up and down depending on what he's playing, but while it's on, it acts as a filter to further alter his distorted sound.
The old chrome Morley wahs show up on eBay often, but they're usually not cheap. There are a few modern wahs which are good alternatives. You could try the current reissue Morley Power Wah, which has a "Boost" knob and sounds similar to the old chrome wahs. The Dual Bass Wah is a bit different and much more bass-friendly, but it can capture some of that essence in Traditional Wah mode. Some people on TB say they can get close with an Ibanez Weeping Demon. And there are other options, too... but you'll need to stay away from the "funky" wahs.