Signs without both will be replaced during routine maintenance following a consultation by the Department for Transport (DfT), the government says.
The plans follow consultation by the Government on its Traffic Signs Regulations and General Directions (TSRGD), which it says is out of date and in need of an overhaul.
In a report, the DfT said: "In order to improve road safety and compliance, ministers have decided that the revised TSRGD will no longer prescribe imperial-only height and width limit signs.
Road signs for bridges, tunnels and narrow roads can currently show measurements in just feet and inches or only metres. Some already display both.
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The DfT said the changes would make it easier for drivers and improve safety.
The DfT said: "Dual measures of height and width restrictions make it easier for drivers and are designed to help improve safety on our roads.
"There are no plans to duplicate any other type of traffic sign."
Britain first laid plans to introduce metric in 1965 but still uses imperial for some measurements.
The government move will anger traditionalists, The Telegraph said.
But some motoring organisations have warned they could lead to confusion. Critics have also expressed concern it could lead to kilometres per hour being introduced on speed signs.
Luke Bosdet, a spokesman for the AA, said: "Motorists and hauliers have got used to seeing signs in feet and inches. In fast moving traffic if they glance up and read the metric metres first, there could be a problem."
"As far as foreign operators are concerned, it's got to be good news as it will make their drivers more aware of height and width restrictions," it said.