Suggestions for the bit before you get changed:
Know your terrain well and can foresee any likely encounters / circumstances that will be complicated if you are messy.
If you don't know the place inside out already, make a survey of least populated or exterior routes to bathrooms, busy/quiet times, most change-friendly disabled cubicles if available, etc.
Practice some methods of avoiding engagement if you encounter someone unexpectedly, that aren't too out-of-character for you.
Plan polite exits if your girlfriend/boyfriend/boss/long lost cousin suddenly rocks up while you're doing a poo and you can't avoid them completely.
Suggestions for changing:
Take all your resources in an inconspicuous bag that you sometimes carry anyway. E.g. I use a laptop bag because I'm always carrying one or more laptops and I might well have one over my shoulder when calling in to the bathroom.
Don't skimp on supplies, in case you have to deal with the unexpected. Anti-bacterial wipes, baby wipes, large tissues and airtight ziploc disposal bags are essentials.
Once you're in the cubicle, there's nothing to worry about, just press on with the cleanup. No-one cares what you are doing, they will all be preoccupied with their own purposes.
Run through some what-if scenarios to reduce the risk of panic if things don't go entirely to plan, e.g. drop nappy wrong-way up on your clothes or door suddenly comes unlocked.
All of this gets easier with practice, but at any time if you're not comfortable with the situation, trust your instincts and don't use your nappy just for the sake of it.
During my 24/7 sessions I'm 100% toilet free at home and at all times for wets. When I need to poo while out and about, I'll do a quick sanity-check and if the risk factors are all low and under control I'll go right ahead and use the nappy. My bottom is quite predictable and I can usually expect a dirty nappy soon after a meal. Knowing all the available options for many of the locations where I will often be during the relevant time period is very helpful. Most often I will set off towards a preferred change location and fill the nappy in response to a natural urge on the way there; if the timing works out I'll be finished seconds before shutting the cubicle door, without encountering anyone else on the way. If conditions aren't favourable, I'll hold it and go on the toilet with I get there.