Can you get brain eating amoeba from picking your nose with finger wet with tap water and soap?

There are no amebae in your scenario. Bacteria can enter the nasal passages though. Using clean water reduces chances of serious pathogenic bacteria from being present. Refrain from picking your nose which can injure nasal membranes making them susceptible to bacterial infections.

There is NO documented case, in peer-reviewed medical periodicals, of an amoebic encephalitis developing subsequent to aggressive "nose picking"! But if you fear contamination of your tap water, you are urged to contact your local water company and inquire.