The 35-year-old, who comes from a wealthy family with royal ties, has lived an isolated life with boyfriend Mark Gordon, 48, since they met in 2016.
The couple and their newborn baby went missing in early January and neither has received medical treatment since.
The family slept rough in freezing temperatures as they tried to avoid being found.
On Tuesday, the Metropolitan Police revealed details of the reward and the last sighting of the family in Newhaven, East Sussex, on Sunday January 8.
Investigators say they took a taxi from Haringey in north London shortly before 1.30am and were dropped off at the harbor entrance at 4.56am.
Police do not believe they took a ferry and instead stayed somewhere in the UK.
The baby was alive but detectives are concerned for his well-being.
Detective Superintendent Lewis Basford, who is leading the investigation, said: ‘We continue to be extremely concerned for the health and well-being of the newborn baby, who has not received any medical attention and has been on the move , and possibly exposed to sub-zero temperatures. temperatures, for almost a month now.
“We are also concerned for the well-being of Constance and Mark, as our investigations suggest they were constantly on the move and awake for at least a few days – which must be particularly exhausting for someone who recently gave birth without care. medical.
“Finding the baby is our top priority and we are now offering a £10,000 reward to anyone who provides information leading to their whereabouts. Perhaps you have already been hesitant to present us with key information for us for whatever reason help find the family.
“No matter why you didn’t speak to us sooner, what matters is doing the right thing now for the sake of a vulnerable newborn baby.
“Perhaps you have seen them wandering the streets? Maybe they asked you for directions? Maybe they got into your taxi or entered your establishment to buy food and drink? Please contact us with all the information you have, as every information we receive will help us find them.
“I want to emphasize that if you have taken money to provide a service to the family, whether it is to provide them with a lift, accommodation or whatever, you have no problem, we just need to hear from you.”
In September last year, the couple began moving to different Airbnb rentals in what may have been an attempt to hide Miss Marten’s pregnancy.
The police search for the family was sparked when officers were called to a burning car on the M61 in Bolton, Greater Manchester on January 5.
It emerged the couple had abandoned the car when it broke down and took a taxi to Liverpool, and from there to Harwich in Essex.
They have since been seen in East London, before moving to North London and then Newhaven.
In east London, they bought camping gear and had a pram, but dumped the buggy at Flower and Dean Walk near Brick Lane shortly before midnight on Saturday January 7.
Mr Basford added: ‘We have no evidence to suggest they boarded a ferry in Newhaven, and we still strongly believe they remain here in the UK – although they may be n anywhere. Our inquiries also establish that the baby was alive and with them when they took the taxi to Newhaven.
“We know they were still in possession of camping gear on Sunday January 8th so I would encourage people to stay alert and watch for a blue tent.
“Please keep your eyes peeled, especially when walking through wastelands or beauty spots, or when walking the dog, and report any information to us.”
Anyone with information about the family’s whereabouts can call the police on 020 7175 0785 or, to remain anonymous, contact Crimestoppers on 0800 555 111 or online.
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