What effects would an electron in the nucleus have?
An electron being at/in the nucleus results in the Fermi Contact Interaction.

This interaction can be observed experimentally through NMR, EPR, and electron capture.  

In NMR of paramagnetic compounds, chemical shifts can be far outside the usual diamagentic range. The contribution to chemical shift of the electron density at the nucleus is referred to as the "contact shift".

For more information on Fermi contact, see "Origin and meaning of the Fermi contact interaction" Theor Chim Acta (1988) 73:173-200.