Installed new Oracle client, now Gateway datasource no longer works.

I'm also an Oracle driver user both with 19c and 12c, so i'm quite familar with your pain.  19c is pretty straightforward at least, unlike the very particular installation you need with 12c where one out of order step can cause you to start again.

Things to check:

1. Make sure you have installed 64-Bit and not 32-Bit drivers

2. Make sure you also installed the 64-Bit ODAC.  It is a sepearate installation package.

3. Check the environment variables (System Properties -> Advanced -> Environment Variables).  In this section look at the 'System Variables' and find the "Path" entry.  Make sure the 19c 64bit Oracle driver is the highest entry compared to any other Oracle installation entry.
4. Still in the environment variables, check if you have an entry called ORACLE_HOME.  It could be pointing to the wrong home folder.  Its possible to have more than one.  Remove this entry if you want to test.  You can always put it back.

5.  Check your oracle installation path for the tnsnames.ora file.  You should find this under  "<Relevant Oracle Home Folder>"\product\19.0.0\client_x64\network\admin

I'd also recommend installing Power BI Desktop on your gateway server.  Open up a project that uses an oracle source.  This should let you troubleshoot faster as you can keep iterating until it works in Power BI Desktop.  Once it works there, it will work under normal gateway circumstances.