I doubt you'll find these films on the internet as they were made back in the days when people rarely put their films online.
You could try the BFI National Archive to see if they have a copy of the films. Other than that, maybe Cambridge Film Festival still have the VHS screener of Larceny that Nolan sent them back in the 1990s. Some festivals keep copies of all the films submitted, even if they are not accepted and people have dug out early efforts by major directors this way. I'm sure Nolan would've submitted Larceny to all the major festivals: Berlin, Cannes, Venice, Edinburgh, London, etc etc. Maybe UCLU Film Society have a copy somewhere or maybe the actors in the film do?
Tarantella may be more difficult. PBS would be an obvious starting point, but it was over 20 years ago and as I understand it, PBS has dozens and dozens of local stations across the US. Nolan's mother was American so maybe it was the local station where the family had connections to the area. (Does anyone know where this might be?)
The obvious impediment you'll encounter is that even if someone does have a copy, why would they send it to you when you might rip it and put it in the internet, which may then mean the copyright owner (ie. Nolan) gives them grief? In my experience, organisations can be very cautious about releasing material to the general public without getting permission from the owner / interested parties. You might have more success if you are an academic seeking to watch the film for research, but I doubt they'll let even them walk away with a copy.
Perhaps the most obvious route, however, is to contact Nolan directly via his agent and ask him if the film still exists and if they are available to view in an archive somewhere. I doubt he'll send copies out to just anyone, but he may be able to furnish some further information. Presumably Cinema 16 who produced the DVD with Doodlebug on contacted him and bought the rights to it - he even did a DVD commentary, but it would seem that Doodlebug was one of his later and more accomplished short films. It may be that he does not want Larceny and Tarantella to see the light of day, but it may still be worth a shot. The most obvious place for Larceny would have been as an extra on the Following DVD - as it does seem to share some striking similarities to that film - but for whatever reason they didn't include it.
Maybe I've missed something, but what is all this talk of "The Dog (2009)"? Did Nolan make a short in 2009? Seems unlikely. Why would he? Can you fill me in?