In reply to JayPee630:
I would be wary of taking them without written confirmation from someone very senior at Eurostar.
Their "dangerous sporting goods" page states:
If you’re travelling with any of these items, you’ll need to contact Eurodespatch, our registered luggage service: ice axes and other climbing and mountaineering equipment, javelins, spears and spear guns, crossbows, crossbow bolts, longbows, longbow arrows and starting pistols, swords or starting pistols. Keep in mind that these are all subject to approval and security checks.
There was a period when thanks to the efforts of the BMC and others ice axes were allowed. Following the terrorist activity in France and Belgium Eurostar have taken advantage of the situation to once again restrict luggage that one can legally take on any other train in Europe.
If you were coming from France then there's a chance that the ice axe would pass as alpinism is a widely recognised activity in France. On the London side there's no chance that it'll pass.
The goal appears to be to discourage people from taking large bags as the trains don't have enough space. After many years of using the tunnel I've now given up.
It's worth remembering that the vehicle services are run by a separate company.