SRTM data is given in Lat/Lon coordinates, while Landsat is given in UTM coordinates. Is there a way to reproject the SRTM data into the Landsat CRS in GDAL (Python)?
The tool in gdal that can reproject your data is gdalwarp.
With gdalwarp, you must provide the source and target coordinate systems with -s_srs and -t_srs, respectively. In your case, it would be
gdalwarp -s_srs EPSG:4326 -t_srs EPSG 326xx your_source _file_name your_dest_file_name, where xx are the numbers of the UTM zone.
Note that you can get the coordinate system automatically if your have images everywhere in the world (see, e.g., GDAL/Python: How do I get coordinate system name from SpatialReference?)