ThermalAlchemy pretty much summed it up above!
However, if you are interested in the original without the 19J6 tubes, the 12AU7 change has been successfully built by many on the forum. I'm pretty certain the 12AU7 version works a bit better than the 19J6 version, simply because the 12AU7 is a much better tube. That schematic, with its very few changes, is documented here:
http://diyforums.org/SSMH/SSMHvariants.php. Given the simplicity of the circuit, changing to another tube does not have a bad effect on sound quality. It might be different if the circuit was more complicated and highly optimized for a single tube type. Personally, I'd settle on the 12AU7 and go from there.
Remember that the entire theme of Pete's original design was
Starving Student. He chose the 19J6 tube because at the time, it was $2 each (even less if bought in volume). Fortunately or unfortunately - depending on how you look at it - the design became so popular that stocks of 19J6 tubes were bought up everywhere within a year or two. Now, it's become a rare tube. No dealer has them in stock in useful numbers (100 or more) anywhere. So, given the laws of supply and demand, the price went up - so much so that the more expensive (and plentiful) 12AU7 became an economic alternative. Use of the 12AU7 is not all because the 19J6 is scarce/expensive; it's also an inherently better tube, period.
The downside is that you are locked into about $30 for a pair of tubes (or much more, if you want to spend it). That's almost the same price as the entire Bill Of Materials (including the original 19J6 tubes) when Pete first design the amp: $35.80 (minus casework).
That said, Pete's new Starving Student design attempts to hearken back to that original premise, with another type of inexpensive tube. You have lots of choices - all of them good.