My father has just passed away. Can the engagement ceremony we planned go ahead?

Yes; the Shariah does not forbid an engagement or marriage during a period of mourning for a family member. For women, mourning (ihdad) for the death of a relative is permitted for no more than three days, except a wife mourning her husband. After that, the decision to go ahead or postpone depends on the family's emotional readiness. Many choose a smaller, simpler ceremony. Note the question of the wali too: that duty now passes to your paternal grandfather or your brother, following the order of wali. Allah says: “Seek help through patience and prayer” (Surah al-Baqarah 2:153). See The marriage wali.