It all comes down to what you consider "conditional". For example, when you trigger an
AND, the output varies depending on its operands: is that a conditional operation?
In this case, Wikipedia is talking about "conditional transport" (the ability to turn busses on and off via a register), not "conditional execution" (the ability to jump to different places based on some condition). The latter is what's needed for Turing-completeness, as you correctly pointed out.
In some TTAs, you can directly write to the program counter; in others, there's a functional unit that handles instruction fetching and decoding and all that, and it has ports for jumps. In the latter case, conditional jumps can be implemented with multiple ports, just like you might expect: write an address to the "jump destination" port, then write a value to the "branch if zero trigger" port, and it'll branch to the destination if the trigger value was zero.
Wikipedia suggests that a single port can be used for this:
TTA implementations that only support unconditional data transports, such as the MAXQ, typically have a special function unit tightly connected to the program counter that responds to a variety of destination addresses. Each such address, when used as the destination of a "move", has a different effect on the program counter—each "relative branch " instruction has a different destination address for each condition; and other destination addresses are used CALL, RETNZ, etc.
This has the same effect: you just have to write the destination and the "opcode" to the same port, one after the other.