The main consequences would be Gore trying to distance himself that much more from Clinton, and thus actually losing the 2000 election in the popular vote as well.
Would Gore have until 2000? If the scandal breaks before the November '96 election (six months pregnant equals summer '96 or thereabouts), would Clinton continue as the Dem candidate going into the elections, or would he resign and let Gore carry the banner? I'm open to the possibility that, given his enormous popularity at the time, Clinton could conceivably (no pun intended) stay in the race and even win. But he would be a crippled president politically for the next four years, especially after photos of the love child started popping up and Monica did Oprah.