This is a question I get often as a Byzantine Reenactor: The simple answer is they weren't.

The problem with early cannons is that during the time of the fall of Constantinople, the interval between loading and firing the cannons was rather long. The Theodosian Land Walls, built in the 5th century AD, were designed with a rubble core that although it made it impossible for them to stand up to modern artillery, they were extremely easy to repair. As a result, during the intermittent barrages, the defenders could repair the breaches using earth and rubble from the destroyed sections. The city actually fell when the defenders reassigned the troops defending the northern sections of the walls after Gustiniani successfully countermined the attempt to undermine the walls, in preparation for the final assault. As a result, when the Turks came across an unlocked postern gate (a gate in the side of a tower used for sallies by the defenders) that had been bricked over centuries ago and then unbricked in the 13th century, they were able to get into the city and there weren't enough defenders in that part of the walls to repel them. As a result morale dropped and the defense against the main assault collapsed, and the defenders either rushed back to try and defend their homes, escape in the harbor, or were killed allegedly fighting alongside the last Roman Emperor Constantine XI.

For a pretty decent general read on this subject you can try Crowley's "1453" which is what I used to reply real quick. There are more in-depth works, such as the peer reviewed "The End of Byzantium" by Jonathan Harris.

EDIT: I see you clarified your question so I thought I'd respond to that. Other forms of siege warfare were used - ladders were used to assault the sea walls from the harbor once they rolled the ships overland on logs through Galata. Ladders are also mentioned in the primary sources in the land assault. Siege towers I think are mentioned as well, I honestly don't know and it wouldn't be possible to get them over the moat unless it was filled in, which would be a difficult endeavour at the least. Siege towers weren't really needed since the majority of the defence was from the middle wall, not the tall inner wall. Undermining was attempted by the Serbians in Ottoman employ, but Giovanni Gustiniani collapsed it with a countermine. Mining technology must have become more advanced in that era because the 5th century walls were impossible to undermine in earlier sieges. Battering Rams may have been used but I've never read anything about them in the Ottoman siege(s), so I honestly don't know.