What is the process and timeline for obtaining a Non-O visa at the Savannakhet consulate?

The short answer is you stay in Lao until the visa is approved.

What you can do to speed the process is

- make a profile on the eVisa online web-portal,

- select your country of passport origin,

- select Savannakhet as the consulate in Lao,

- select "stay with thai family more than 60 days" choice under Non-Imm visa

- upload ALL your documentation then save the application in your profile

The second you get to Lao, take a full page photo of your Lao visa/entry stamp, upload it as proof you're in Lao, then submit the application. They're open M-F 9-11 for payments, just show up and pay.

Then go to the consulate with the print out of the confirmation of application and pay the 2000baht for the visa

You SHOULD be approved by the following business day afternoon

That's about all you can do to "speed the processing up"