Alex Neill, chief executive of consumer resolution service Resolver, said UK tourists needed to be aware they were "taking a risk".
Ms Neill, who was previously a managing director at consumer group Which?, said: "While the Government is focused on getting the travel industry and the economy moving again, it's the consumer who is bearing all of the risk.
"It is likely that new travel insurance policies won't cover you if you develop symptoms before going on holiday - and if the flight or packaged holiday isn't cancelled then you have no automatic right to compensation."
Rory Boland, editor of Which? Travel, said: "Holiday companies and airlines should ensure they continue to offer customers flexible rebooking options.
"If the Government is to get people travelling again successfully, it needs to restore confidence in the sector by providing support for the industry and working with the regulator to ensure companies are abiding by the law on refunds for cancelled travel."
According to figures from the Association of British Insurers (ABI), in more normal times, insurers deal with around 1,300 travel claims typically every day.
People can run up huge medical bills for conditions other than coronavirus.
Large medical bills paid by insurers in recent years have included: £241,000 to cover heart problems suffered by a holidaymaker in the United States, £200,000 to treat a brain haemorrhage in China, £153,000 to treat a fractured arm in San Francisco, £89,000 following a heart attack suffered by a tourist in Turkey - underlining the importance of having adequate travel insurance in place.
People who are unsure what their insurance will cover should contact their insurer, as the terms of policies vary.
A spokesman for the ABI said: "Provided it is deemed by the Government safe to travel to your destination, you will be protected by your travel insurance, including cover for any emergency medical expenses. For current travel advice check on: www.gov.uk/guidance/coronavirus-covid-19-travel-corridors.
"Most policies taken out or renewed after the pandemic was officially declared are likely to exclude cancellation due to coronavirus as it is a known risk, and travel insurance is designed and priced to cover you against the unforeseen."
Graeme Trudgill, executive director at the British Insurance Brokers' Association (Biba), said: "Travel insurance is in fact quite complicated, and at this time it is more important than ever to make sure you get it right.
"A specialist broker can help you get the right cover and Biba can help you find a broker at www.biba.org.uk.
"If you already have a policy, it would be a very good idea to check the wording to see what is and isn't covered and speak to a specialist broker about what cover is in place for Covid-19.
"Apart from the challenges of Covid-19, you may have specific requirements, such as making sure a particular medical condition or disability is catered for, or that suffering an injury because of a sport or activity while away is included. Many specialist brokers can help you make sure nothing is overlooked."