Brocolt: With the low cost of a Lee Load-All, the simplest choice would be to purchase one of each. That way, you have a dedicated press ready to go for either gauge, with no additional set up. Nothing wrong in purchasing a conversion kit, though. If that suits your budget and shooting needs better, then go for it.
A 12-gauge press would be the obvious place to start. Then, add the 20-gauge conversion kit, if necessary.
As others have already mentioned, if you only plan on shooting a limited number of 12-gauge shells per year... particularly for limited casual shooting, there are no real cost savings. The real benefit comes from being able to tailor loads to your specific needs..ie: Turkey hunting. That's where reloading really shines. Or, if you plan on shooting competitively. Where going through hundreds, or even thousands of rounds per year, can be common. Then, the cost savings do add up.
There are savings with 20-gauge. Again....depending on how much you plan to shoot. Plus, the ability to tailor loads to your specific needs and gun.
Though the Load-All is a good press(Have owned one for 4 decades and counting)..... it does have it's limitations. The lack of press adjustment being the biggest issue. Though really not that much of one, in my experience. Or insurmountable. Mostly a matter of paying attention to the load column. For consistent, tight crimps. Which many shotgunners...myself included, regard as being one of the most critical components for good shot shell performance. Not hard to do.
I should mention that Mec presses (which are adjustable) will need some fine tuning as well, to attain good crimps.
Still, the Load-All is a good, basic reloading press that turns out good quality shot shells. Plus, capable of delivering many years of dependable service. A great bargain, in my view. Purchase one with confidence and start reloading.
I fully agree with others though, who recommend purchasing a Mec loader. A Mec 600, for example, is always a good choice. Very rugged. Plenty of adjustment and easy to set up and run. Produces good quality shot shells. Pricier than the Lee.....but, a great long term investment. Well worth the cost.
Have run a Mec Sizemaster for more years than I care to count. With thousands of rounds going through that press, without a single hitch. That press will easily outlive me.
Worth mentioning: Mec also offers excellent customer support *. Parts are readily available. Mec even features a series of online how-to videos, on proper set up and running of their product. Another handy resource.
Well worth a look, before making up your mind.
In any case, hope some of this advice is useful.
Al
* Lee does, too.