Between Shoreditch and Aldgate the A1202 forms part of the London Inner Ring Road and the congestion charging zone boundary.
It starts by branching off the A5201 Old Street to the right, becoming Great Eastern Street, with the B101 Pitfield Street as an even sharper left, or a slight right if coming the other way. Curtain Road will cross us shortly and this road forms the diversion for traffic accessing the A10 Shoreditch High Street, which we cross shortly after but has some strange traffic restrictions for northbound traffic. The whole thing used to be a one-way gyratory and they changed it to make all these roads now two-way but a part of Shoreditch High Street northbound is for bus and cycle only.
After this junction the road name is Commercial Street which runs parallel with Brick Lane for a while, with Wentworth Street being one of the roads to the right; this is also known as Petticoat Lane market, thus probably leading to the name of this road (which should not be confused with Commercial Road, which we will almost meet later). We finally arrive at Aldgate, a junction with the A11. This junction has changed recently as it used to be also a big roundabout / gyratory with Whitechapel High Street being one-way East only and Braham Street one-way West. Now, however, it is two-way. Leman Street is opposite and provides the continuation of the A1202 and traffic wishing to avoid the congestion charge must be certain not to miss this junction now. So as the A13 Commercial Road is slightly to the East, it does not actually cross the A1202 directly any more.
The ring-road itself takes a confusing twist here, and we cross a couple of B-roads: the B134 Alie Street here (one-way only to the East; it then heads north across both the A13 and A11 to become the aforementioned Brick Lane) and the B126 Cable Street (which runs along the DLR for a bit and directly between the A1203 The Highway and the A13 Commercial Road and it is really no more than a rat-run between those two main roads so it is classified for historic reasons only).
Your safe bet for Tower Bridge if you don't want to get caught up with any one-way streets is to remain southbound on the A1202 through Dock Street until we hit the A1203 which is briefly East Smithfield in the short stretch to the west (right) and The Highway to the east (left). Here the A1202 terminates but turning right will take you directly to Tower Bridge.