Bus vs Signal Harness in Altium
You're using harnesses in a slightly strange way - not wrong as in "it will work and at least the signals are grouped" but what I would do is to do the following:


Learn the difference between a harness type and the harness name (net label put onto the harness). The harness type is e.g. UART, it will have four signals named RXD, TXD, RTS, CTS. Think of object oriented software. You have a type (class) and you generate multiple instances (UART1, UART2, UART3, ... all of them are of type UART).
With that knowledge, separate the 3 harnesses from your picture; all of them are of type UART (same type) but you apply different net labels to each of the harnesses  (UART1, UART2, ..).
Now, automatically you will get the following signal names in the PCB: UART1.RXD, UART1.TXD, ... You'll also get the correct net classes if you have set the option in the project options


You probably weren't really using busses because as you state, the signal names for your bus must be somewhat sequential and RX/TX do not follow that. As you said, AD probably connected your net labels through the "global" setting which I would recommend against.