Based on CAA(UK) guidance, the presence of a varicocele or hydrocele does not seem to be disqualifying. However, there is no specific guidance on these conditions which makes it harder to asses.
Moreover, which operation does your doctor recommend you have done for your varicocele? You can't really 'drain' a varicocele, you could have an open operation where an incision is made around the base of the scrotum and the vein that supplies the side of the scrotum in question is ligated (tied off). This is a relatively minor procedure that (barring any complications) looking at the CAA guidance, would possibly incur a shorter period of unfitness postoperatively but once assessed as fit again by your AME, should probably mean relatively rapid reinstatement of your fitness to hold a Class 1.
The other treatment option which I would imagine is more commonly done nowadays, is so-called embolisation, where little metallic 'coils' are inserted into the vein that drains the left side of the scrotum (which is the affected side in the vast majority of cases). This is done to obliterate the vein and thus reduce the swelling caused by the varicocele. The procedure is done under local anaesthesia on a day-case basis and I would imagine that if well postoperatively, fitness to fly would not be affected beyond the first few days. However, I cannot seem to find anything about this type of procedure in the CAA guidance material so I would suggest getting in touch with your AME.