If North America were in the southern hemisphere, would the mountains and high desert have to be -east- of Tornado Alley?
No, the mountains would still need to be west of tornado alley.

It all starts with temperatures.
You often hear the colloquial "warm\cold airmass battle" image.
But more specifically, the vital requirement for tornadoes is warm air advection.  Why?  Because warm advection fundamentally means the proper wind shear (cyclonic turning with height) for sustained storm rotation [for more info on why warm air advection is needed, look into thermal wind].
To have winds that bring warm air, you have to also have periods with cold air advection too.  Plus the oscillations between warm air and cold air give the potential for uneven vertical temperature profiles to develop, creating instability to fuel vigorous storms.  Warm air advection also builds the instability by transporting in that warm and moisture low level air.
And then fronts between warm advection zones and cold advection zones brings the trigger to best set off those ripe, unsettled conditions.  So on the warm sides of fronts is the prime location for supercells, though anywhere in warm advection has some potential.

Where do you find this battle of warm and cool air masses?
In the midlatitudes.  The most favorable tornadic regions have to be there, it's the only place with such regular oscillations.

And then the most favored spots within that band comes down to the positioning of mountains.
Mountains help pool the ingredients, and allow upslope flow to perhaps help kick off some of the storms.
But most importantly, mountain ranges perpendicular to the flow cause consistent lee troughing, the formation of surface lows downwind from the mountains when upper-level storm systems approach.

So it comes back to hich way the winds move in the upper levels of the troposphere in the midlatitudes.

And in the southern hemisphere, just as in the northern hemisphere, flow is mostly from the west at midlatitudes [for more on this, look into global or Hadley circulations, particularly the Ferrel cell].

So... indeed, we're talking east of mountains, since the wind is coming from the west.

Lee troughing is really important as young storm systems have strong meridional wind ahead of the system (bringing that juicy air in), the largest warm air region size near the low, and falling pressures, which further enhance low-level wind shear [look into isallobaric flow].  Areas further downstream (east) will have more mature systems with less warm advection collocated with instability.  That's a primary reason why the central US has more tornadoes than the eastern US.

Then it's all about large bodies of warm water to the right of the upper-level wind, such that it brings in that fuel in when a storm system starts to develop.  Because while warm advection is vital for the wind shear, you still need the moisture for tve precipitation (and to really ramp up instability).  The body of water needs to be warm (to evaporate lots of that water, and help that advection have that vital warming quality) and generally quite large (to provide lots of fuel regardless of slight wind differences, and do so regularly)

Finally, you did well to mention the high deserts\plateaus, as that is a pretty special ingredient to the Southern Plains.  As you may well know, a cap is a pocket of warm air aloft that prevents storm formation.  It's usually up around 850-600 mb, which is up a km or two.  High enough to be above  cloud base but low enough to counteract the nascent updraft energy... and also low enough to be driven by  ground features in other places.
That cap prevents widespread storms as you said, allowing a rotating storm to mature and persist.  It's a subtle ingredient in that you need just the right amount of it to keep excessive cells down, but not too much such that it prevents all storms.
It isn't nearly as vital as other features.  But it certainly helps.
And as you hinted, the best source for a cap is a dry hot elevated region upwind at that wind level (which works out to be to the southwest in the Northern Hemisphere... such that the cap in the southern Plains is primarily due to the Mexican mountains, and in the central Plains based more on the US desert southwest).
Even without such a region, there can still be natural capping from the mixed layer of the previous day, though.  And even widespread convection can be prolific at tornado producing if shear and instability are strong enough (see April 27th outbreak for instance).  So this isn't as critical as other ingredients.

So again, the order of most important to less important are:
1 - warm advection (= midlatitides, in spring\fall)
2 - Downwind of mountains (= east since midlat flow is from west)
3 - Warm moist water to right (south in NH, north in SH)
4 - Deserts upwind of capping level wind field (SW in NH, NW in SH)

The best spot is a reasonable balance between each ingredient, though still quite close to each source.
And that's all the Southern Plains of the US.  Out Oklahoma way.

One thing that actually turns out not to generally be as important is the geography in the cold air direction.  I've definitely heard that one suggested before, but don't see it factoring in much.
While the amount of land Poleward can help keep cold air drier and allow it to get colder perhaps overall, it's really not the strength of the cold air as much as the existence of it.  Too strong of cold and dry air and you'll have it reaching the warm water source, and disabling moisture return for long periods of time.
That's why tornado season in the US tends to peak a bit after warmth starts returning.
You can look at where prolific tornado areas are, too... the Great Plains have a lot of Poleward land; so does a lot of Eurasia, yet it's mostly unfavorable.
And then conversely, Australia doesn't have a lot of land Poleward yet is still a reasonably busy tornado spot.

Mountain ranges Poleward also generally aren't a big deal... even pretty large ones; one of the biggest tornado areas is Bangladesh\India, despite the disruption the Himalayas presents.  Because enough cold air can filter into the region in spite of the blockage.

The only thing that would be a real large scale geographic issue to the needed cold air would be a warm sea Poleward that modifies the incoming cold air significantly.
But that's pretty tough to have geographically.  Perhaps more possible in the Autumn... maybe the Great Lakes serves as a slight dampener on fall season tornadoes in parts of the US?  Though their effect is still mostly pretty small overall.  Bring too near any large body of water, in any direction, is really a downer on supercellular tornadoes, as it modifies temperature gradients and instability.

To your direct question... there actually is a very good example of what Tornado Alley would look like in the Southern Hemisphere already: the Pampas Lowlands of Argentina.
It has that big mountain range to the west, in the midlatitudes, and does have fairly warm water a ways northward.

But it doesn't have quite as large of a region east of the mountains to have the tornadoes in, the cold air (and storm system strength) is probably modified due to the closed nature of the Antarctic vortex and the widespread oceans of the SH modifying air masses, and the source water isn't probably as well located being so far north (and is it warm?).
But even still, Pampas area might be the second most consistent tornado region on Earth.  Such that some US storm chasers have traveled down there in our winter.

In the end, local effects play a huge role too, creating mesoscale ingredients (seabreezes\temperature boundaries, upsloping, local vortex flow, etc) to add plenty of rotation to the ledger (see Florida, one of highest tornadoes per square mile in US, in large part due to seabreeze water spouts and hurricanes).  But all things equal, being east of mountains in the midlatitudes is a jackpot ingredient to climatological tornado formation (regardless of hemisphere).