Exact mechanism for activating Order 66 protocol in inhibitor chips is not described either in Canon or Legends. However, we know few things: it was activated by voice command, presumably over encrypted secure channel. Legends assume it could be activated only by Chancellor (Palpatine) although there is no confirmation for that in Canon. Palpatine, of course, could not contact each and every Clone commander personally, and each and every clone even less so. Therefore we could assume Order could be relayed down the ranks, although exact mechanism, as mentioned before, is not known. Somehow, effects of Order 66 were immediate . For example, Commander Cody, after receiving Order, ordered unknown trooper(s) to fire at Obi-Wan and was obeyed instantly.
As for Cut Lawquane, we know he was not inside GAR structure when Order was issued. I doubt Palpatine knew or cared about him, and would not have any reason to contact him personally. Cut didn't have GAR communication equipment with him, and was not part of any GAR communication network. He didn't wear standard issue helmet, so Order could not be transmitted to him that way. Only other way remaining would be if inhibitor chips themselves acted as receivers, but I doubt that, otherwise Palpatine would not need to use holoprojector to contact Cody and others.
Therefore, most likely, Order 66 protocol was not activated inside Cut Lawquane's inhibitor chip. He would not attack Jedi if he saw one . Although Order 66 most likely remained standing for years after it was initially given(to hunt down surviving Jedi), as far as we know it was not repeated. So, clones not affected by it immediately were not affected at all.