Live updates: follow the latest news on Russia-Ukraine
The crossbench peer and former head of the British Army was speaking to BBC Radio Two’s Jeremy Vine show.
He said he had spoken to officials himself to progress cases for a group of six people heading to his community, two of whom he said he hoped to sponsor.
“Whereas other European countries have open arms, welcomed very large numbers of refugees into their country, we are applying a quite complicated, a very complicated visa system,” he said.
“We’ve designed a perfect system, but actually it’s so complicated that it’s imperfect.
“And we do need to be more generous, we need to be open-minded. We have to be prepared to take a few risks in order to get people to a place of sanctuary and safety.
“And it’s a challenge to the government. I mean their heart’s in the right place, but they have made it so blooming complicated that we’ve got to find a way of simplifying this, and sharpish.”
“There was an option available to members of the House of Commons and House of Lords to come to London and to meet Home Office officials and check up on the application process,” he told the programme.
Lord Dannatt said he had taken up the offer and visited on Wednesday, “and lo and behold, of the six applications that I’ve been tracking, three, as of this morning, have now been granted, and I’m on the case of the other three”.
“Everybody who is sponsoring someone to come across from Ukraine has got a Member of Parliament. So raise the case, if it’s not going fast enough, to your member of Parliament, and your Member of Parliament can do what I’ve done this morning, which is get in touch with the Home Office directly, come to Portcullis House where you can meet officials, and that case will be actually effectively fast-tracked. But every case can be fast-tracked,” he said.
Lord Dannatt called for more Home Office staff to be assigned to the task.
“My plea to the Home Office is put more staff on to this, simplify the process and try and expedite it as quickly as possible,” he said.
“We have devised a very complicated system which does need simplifying. I mean, I know we’re trying to protect our borders and all the rest of it, but we’ve made it too difficult. The application form is nine pages long.”
A caller to the programme described the system as “horrific”.
Lord Dannatt also called on local authorities to be “sensible and pragmatic” when assessing the accommodation people are offering.
“If we want to be genuinely welcoming as the United Kingdom, then yes, OK, a visa process is what we’ve chosen to do, but let’s make it simpler, let’s make it quicker, let’s have more Home Office staff processing these applications and get people here,” he said.
“And local authorities need to be sensible and pragmatic about approving the accommodation which people are offering.”
The Homes for Ukraine programme was launched on March 14 with the aim of allowing individuals, charities, community groups and businesses to bring Ukrainians, including those with no family ties to the UK, to safety.
About 12,000 people had arrived under Ukraine visa schemes as of April 5, according to Home Office figures.
About 10,800 people had arrived under the separate Ukraine family programme, but only 1,200 had made it to Britain under Homes for Ukraine sponsorship, provisional data published on the department’s website showed.
“We continue to process visas for the Homes for Ukraine scheme as quickly as possible, but accept progress has not been quick enough,” said a government representative.
“The Home Office has made changes to visa processing. The application form has been streamlined, Ukrainian passport holders can now apply online and do their biometrics checks once in the UK, and greater resource has gone into the system.
“A UK Visas and Immigration helpline can provide information on eligibility and applications, and in cases of concern can escalate to teams who can look at the full case history and establish any issues.”