Can I travel within Europe, while both my current and new passports are in the post or do I effectively have no passprt during that time?
If you have a passport card, or another travel document such as a foreign passport or a laissez-passer, then you can continue to travel on that.
If you do not have any kind of travel document in your possession, then in practical terms you cannot exercise your right to travel in the European Union until you receive your passport.
If you are a British or Irish citizen you can still travel from Ireland to the United Kingdom, the Isle of Man and the Channel Islands without a passport. If you opt to travel by air or by boat, you may need to present alternative identification such as a driving licence. None of the UK/IOM/CI perform immigration checks on arrivals from Ireland (although they may do random spot checks very occasionally) but Ireland does check ID at major airports and ports---a driving licence will be sufficient to re-enter. Your travel carrier may impose their own requirements as well.