A visit to Morocco by the Egyptian dictator Colonel Gamal Abdel Nasser had convinced many Jews that an ill wind was blowing through the Middle East and that they were no longer safe. "He left his family and went on a boat, with no papers, no belongings, no passport," Mrs Lawrence said.
Taken to a refugee camp in Marseilles, he eventually made his way to Israel, going on to fight in the Six-Day War.
When Mrs Lawrence saw pictures of boats landing refugees on the shores of Greece last September, she felt that "the images were directly comparable to the story of my father. I felt compelled to help people as my father had been helped."
Her mother is an émigré from North Africa, too, from Algeria. Born in France herself, Mrs Lawrence recalled that when she saw in the media the conditions endured by the refugees in Calais, "being French, I was so ashamed of what was going on".
Her first instinct was to fill up a car with goods and drive there herself - which she said now "is the worst thing you can do because you can't just turn up with items you haven't checked are needed."
Having got to hear about an initiative to collect £1,000 worth of items and send a van, she joined forces with two of the women who were to become directors of Help Refugees, Josie Naughton, and Lliana Bird, a radio DJ who is also Jewish. Within a week, a social media campaign had raised £56,000 and hundreds of goods were pouring in.
"We knew in Calais they needed new men's boots, warm clothes, sleeping bags, tents… We made an Amazon list and started receiving 7,000 packages a day from all over Britain."
The next task was to transport the contributions. "For five weeks, every morning, I'd drop my kids off at school and at 9am I'd be at the Big Yellow Storage facility. We asked people to come and help and hundreds did every day… Nandos and Dominos gave free lunch to keep these people going."
When they went to Calais to work out the logistics, they discovered there was no warehouse. Within five days, they had partnered with a French charity, L'Auberge des Migrants, to find one to use as a distribution point. The first convoy of 19 lorries - provided by Tesco's - set off.
She said: "The thing that shocked us most in Calais was there were no large organisations there. There were volunteers trying to help, but no organisation on the ground at all." Now, the warehouse has a kitchen and cooks meals cooked for an estimated 6,000 refugees in Calais and Dunkirk. The charity has also built shelters to house refugees.
But that was not enough. When Mrs Lawrence saw the pictures of the perilous boat journeys to Greece, she and others felt they had to extend their operation. They funded a Bristol-based organisation called Skipchen to set up a mobile kitchen off the Greek coast.
"We started to support search-and-rescue on the shores of Lesbos. We fed the people who came off the boats and helped process their [asylum] applications."
Now they are supporting 27 camps on the borders of Greece. "The Greek government are allowing Help Refugees and other NGOs to service these camps because they are unable to," she said. "They don't have the funds."
They are also trying to help an estimated 95,000 unaccompanied refugee children. As many as 10,000 are thought to be lost somewhere in Europe. "It is the same figure as children brought over in the Kindertransport," she said. "We are determined to do everything we can to get those children out."
Help Refugees has backed a campaign to bring at least 3,000 unaccompanied children to the UK immediately.
One seven-year-old boy, Ahmed, from Afghanistan, who had been in Calais, managed to stow away with 14 other refugees on a truck. But when they arrived in Kent, they found themselves trapped in a sealed container with the air running out. On a mobile phone given to him by Help Refugees, Ahmed sent a text to Liz Clegg, a volunteer at Calais. Although she was in the USA, she was able to alert Help Refugees' head of communications, Tanya Freedman, who called the police just in time.
"When we first started, our urgency was based on keeping people alive and providing basic humanitarian aid," Mrs Lawrence said. "As time has progressed, it has become so much more. We are trying to get the children out. They are vulnerable to being abused and trafficked. There are stories I can't tell publicly."
The charity, she said, has "been overwhelmed" by the level of public goodwill. "We see our role as facilitating others to help on the ground."
A Syrian refugee, who has been able to come to Britain, told her: "I can't believe you are a Jew helping an Arab. I have learned so much from this."
For the past eight months, she has had to leave the toiletries business she helped her husband Dominic to run. "He's been to Calais with me, he understands the need to help."
This week, she went to a camp at Idomeni on the Macedonian border, where "conditions are absolutely shocking".
While some have sought to draw a distinction between economic migrants and refugees from war zones, Mrs Lawrence is in no doubt of the desperation that has driven so many to undertake their journeys. "Nobody who is nine months' pregnant would get on a boat when her waters might break unless they feared for their safety," she said.
