On Friday night, the British government told its citizens in the country to “leave now while commercial means are still available”.
While some have vowed to stay, others are now trying to work out how to get to the UK as quickly as possible.
Ben Garratt and his wife Alice, who live in north London, have been in Ukraine since December. Their baby was born in the country under its “very different surrogacy laws”, which allow for a swifter IVF and surrogacy process.
They are now waiting for their son’s emergency travel documents to come through before being able to leave the country in line with the UK government’s advice.
“We’re essentially stuck in Ukraine until we can get the required British paperwork to allow our son, [who was] born here, to leave,” Mr Garratt said.
The 40-year-old is due to have an interview with the passport office in Kiev next week to provide proof of his son’s British citizenship. “They want to do a two-hour phone interview with me, but that’s not until Wednesday,” he said.
Stuart McKenzie, originally from Scotland but now living in Ukraine, said he was worried about what would happen as potentially thousands rush to leave around the same time.
“With young children in the country, I’ve got to take their safety as a priority, so we’re definitely looking at how to get them out as soon as possible,” he told BBC Breakfast on Saturday.
“So many people are trying to leave at the same time and there won’t be flights, the roads will be blocked, are you going to be able to get fuel for your car? Is there going to be cash in the banking machines?”
The businessman added: “There could be so many things happening, so much chaos happening.”
Other Britons in Ukraine posted on social media saying they would stay in the country as it faced the threat of an attack by Russia. Some said it was not easy to leave immediately because of family ties.
British journalist, Tim White, who is currently stuck in Ukraine’s capital, followed the UK government’s advice to leave the country and booked onto a flight with Dutch airline KLM due to leave Kyiv on Sunday. Hours later, however, the service was cancelled.
He has since been able to reserve a seat on another flight with a separate airline, but added in a Tweet: “If that falls through I’ll be driving or train and hoping borders stay open!”
On Saturday, the UK’s armed forces minister said all British troops would be withdrawn from the country this weekend, as he warned that an invasion could happen “at no notice”.
The UK government says that Russia has amassed forces on Ukraine’s borders since last month, which has increased the threat of military action.
Russian troops have also been sent to neighbouring Belarus to take part in exercises. Russia denies it intends to launch an offensive against Ukraine.
Additional reporting by Press Association