Sir Ed set out his argument in a speech to his party's spring conference in York on Sunday, saying the UK's continued reliance on US support is an unacceptable risk to national security.
Sir Ed told the conference: "Britain's nuclear deterrent must be genuinely, verifiably ours – not dependent on Trump or whoever his successor may be. Trump has proven we can't rely on America as a dependable ally.
"As the UK now prepares to replace Trident in the 2040s, we should make the decision now to spend the billions required over the next two decades here in the UK, not in the US.
"Britain has the best scientists, the best engineers, the best builders in the world. So let's get building our own, truly independent nuclear deterrent here in the UK."
Speaking to the BBC's Sunday with Laura Kuenssberg programme earlier, Sir Ed added: "If we start now, we can be in a place where we control the missile technology ourselves and spend the billions that that's going to cost not in the United States but here in the UK."
In response a Ministry of Defence spokesperson said: "Our independent nuclear deterrent protects us every minute of every day and is the ultimate guarantee of our national security."
They added: "As the 2025 Strategic Defence Review made clear, a modernised nuclear deterrent will remain the cornerstone of the UK's defence, and our commitment to Nato and global security."
The Lib Dems have not provided an estimate of the cost of their proposal, which would far exceed the billions being spent on the planned replacement for Trident.
France, the only other European country with nuclear weapons, has always maintained a fully independent system.
The Lib Dems say France's approach proves a sovereign British capability is achievable.
The French missile programme in the 1970s is estimated to have cost about £20 billion in today's money and the party says this cost would be split over two decades.
They argue it could be done in two stages - developing a way to maintain the existing Trident weapons system domestically, and in the longer term manufacturing a fully British-made replacement.
The party has traditionally been split on the issue of Britain's nuclear weapons.
Many in the party have campaigned for multilateral disarmament - something party sources insist they still believe in, while saying they must reflect the reality of the global situation.