Had the Allies hit along the east coast of Italy, where the British V Corps were the same would have probably happened up that coast also. After all, there's only two routes in Italy heading north, Route 6 and Route 7.
Route 6 was the aim of the capture of Cassino, and Anzio landings. And Route 7 was on the east flank of Italy.
To take Route 7 the exact same would have been needed as to take Route 6. Only then it would have been a different mountain that became famous.
It took something around six Corps to eventually break on to Route 6, if I remember correctly. I highly doubt that British V Corps could do it on Route 7 themselves. However, even if the Allies moved in reinforcement to the British V Corps to break that route there's no way of them getting tanks over the Appennines to Rome until they get far up north. And that would mean the left flank, U.S 5th Army, would still be stuck at Cassino.
That is to say that British 8th Army could have even broken through on to Route 7. Remember that the Gustav Line spread across the entirety of Italy, that's over Route 7 too. The line wasn't just a static line, it had 10th Armee mobile reserves, plus in 1944 the Germans received two fresh divisions from Russia. Had an attempt been made on Route 7 they would have rushed their reserves that way instead of to Cassino.
Now, my thoughts are, beside just giving the FEC a reserve to let them break to Atina in January 1944. Would be to land at Anzio then have them breakout while Route 6 and Route 7 was attacked at the same time. The German forces wouldn't know where to throw reserves and at least one attack would have broken through. But that's the benefit of hindsight.
The reason the Allies failed so many times was because they were throwing in little bits of their strength on a narrow frontage each time. This allowed the Germans to pinpoint the attack and bring it under artillery fire and send all their reserves to that spot to counter-attack. A broad frontage of attack en masse would have broken the enemy.