There are two solutions that come to my mind, one more HW based and another more SW based.
The HW way is to let the programmer control the LCD through 2:1 multiplexer(s). You will need to connect the LCD pins to the common pin of the multiplexer, the Programmer's ones to one of the two inputs and the Worker's to the other. Then the programmer can choose who controls the LCD through the additional pin. An example of the multiplexer you can choose is the 74LS157, but there are virtually infinite part numbers. The disadvantage of this approach is that you need to write the whole LCD control stack in the Programmer.
An alternative, more relying on the SW, is implementing a way to let the slave know that you want to program it. For instance, when a programming is needed the Programmer raises a pin; the Worker writes the string on the LCD, then stops working, raises another pin and waits to be cleared. The programmer then proceeds to flash the Worker, and the message will remain on the LCD. It may happen that garbage exits from the Worker's pins when being programmed. If so, change the enable pin or find a way to temporarily detach it from the Worker to avoid signals reaching the LCD. This schematic illustrates this concept:
simulate this circuit – Schematic created using CircuitLab
When the Programmer wants to program the unit, it sets the DL_Req pin to high. The Worker sees this, and performs accordingly. I think pins are not moved then, but if for instance some noise appears on the signals during programming (and the LCD displays this noise as weird chars), try changing the pin associated to the enable pins (LCD2 in the example above) or put something to prevent noise on that wire only, so that LCD content is not modified.
Personally I'd go with the second solution, but this depends on how much you can change the behavior of the worker and programmer