That's exciting that you've got a plan in place Pickle 
Really don't worry about your periods. TMI coming up... Mine had dwindled to only one day of bleeding heavily enough to use a tampon then a couple days of very light bleeding (needing only a panty liner). It doesn't seem to have caused problems for me.
The thing that is important is that you have at least ten days (I think the average is 14 days) between ovulating and the first day of your period. Its called the luteal phase and is the time an embryo would be given to implant, so the longer the better really!
I'm assuming that Kwick is hibernating, which is what I would be doing. But if you're reading the thread please know that you cross my mind every day and I'm trying to send supportive vibes in your direction 
Karen - crossing my fingers for you. You sound like you're in a good place
I am not generally a dog person
but your puppy is absolutely gorgeous! That photo lying on the bed 
Caution - re Australia, I've been thinking about this quite a bit and I really think you have two options.
a) you put off your first IUI (and all the preceding tests) until after the trip to ensure you're not pregnant when you go
b) you don't go
Once you do start the fertility treatment process you will need to give it your all and be whole-heartedly invested in it (not dealing with an underlying concern about what each outcome means for the Australia trip).
You would be foolish (sorry) to make any decision based on a calculation of what stage of pregnancy you might be at by the time of the trip. Enough of us here have proven that things rarely go to plan and that it generally takes a number of attempts to get a BFP and that even then holding onto those BFPs isn't a given.
I would have been physically fine doing a long-haul flight at 23 weeks. Now at 28 weeks, I was really uncomfortable just having to sit at my desk all day in the office on Monday.
Also, at the point a baby would be viable if you delivered early (23 weeks+) I wouldn't want to risk being in Australia. Imagine dealing with a premature baby in NICU in Australia (and the hefty travel insurance you would need to take out to cover this potential eventuality) and the many months you would have to wait there until your baby was fit to fly. 1 in 9 UK babies are born prematurely and require specialist hospital care, so although its unlikely, it could happen.
Sorry to be so negative but you need to work out what you want. If you would absolutely love the Australia trip and can come to terms with leaving getting pregnant until afterwards, then do the Australia trip and enjoy it! Just don't do it to please other people - a course of action which rarely works out best for everyone in the long run.If you are desperate to get started before the Australia trip then I think you have to do what is right for you. This is too big a life decision to be worrying about letting others down.