The 5% tax on all Mexican goods was supposed to begin Monday, but Trump reached an agreement with Mexico to reduce illegal immigration.
But farming, car industry and ceramics will be protected from cheap imports, ministers say.
But minister says measures would only be implemented if Britain was on course to crash out of EU.
"In a stroke of a pen, the Department of International Trade could kill off one of our oldest and proudest industries."
Environment secretary confirms UK will impose tariffs on EU imports in bid to protect farmers.
But cabinet colleague Greg Clark said government must "defend" industries from foreign competition.
We asked 36 people including a solicitor, an engineer and an accountant, to work out an energy bill using nothing but information from a supplier's website. Just one - a company director - could do it.