The insects which spread "vector-borne" infections have already begun to invade parts of southern and eastern Europe.
The UK climate is already said to be suitable for mosquitoes carrying West Nile Virus, an infection that causes a flu-like illness which on rare occasions can prove fatal.
No human cases have come to light so far. But recently a species of Culex mosquito known to be the main carrier of West Nile Virus in Europe was discovered in Kent.
A bigger health threat is the Asian tiger mosquito (Aedes albopictus) which spreads dengue fever and chikungunya.
Both cause serious illnesses which, especially in the case of dengue fever, can be fatal.
Warmer temperatures and more rainfall - both outcomes of climate change - could provide ideal conditions for the mosquito in the UK, particularly southern England, said the scientists.
Previously, dengue transmission was largely confined to tropical and sub-tropical regions because frosts kill the insect's larvae and eggs.
The Asian tiger mosquito has now been reported in 25 different European countries and is widely established in large parts of the Mediterranean.
It has also been imported into the Netherlands, where the insect has been found in commercial greenhouses and its eggs and larvae in water trapped inside the rims of used tyres.
Climate change models show that just a 2C rise in temperature could extend the mosquito's activity season by one month and geographical spread by up to 30% by 2030, said the researchers writing in The Lancet Infectious Diseases journal.
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Professor Steve Leach, from the emergency responses department at Public Health England, said: "We are not suggesting that climate change is the only or the main factor driving the increase in vector-borne diseases in the UK and Europe, but that it is one of many factors including socio-economic development, urbanisation, widespread land-use change, migration, and globalisation that should be considered.
"Lessons from the outbreaks of West Nile virus in North America and chikungunya in the Caribbean emphasise the need to assess future vector-borne disease risks and prepare contingencies for future outbreaks."
In the past decade insect-borne infections have spread into new territories across Europe. Examples include malaria in Greece, West Nile Virus in eastern Europe, and chikungunya in Italy and France.
The climate change simulations predict suitable temperatures for one month per year of chikungunya virus transmission in London by 2041, and up to three months in south-east England by 2071.
Co-author Dr Jolyon Medlock, also from Public Health England, said: "Given the on-going spread of invasive mosquitoes across Europe, with accompanying outbreaks of dengue and chikungunya virus, Public Health England has been conducting surveillance at seaports, airports, and some motorway service stations.
"Although no non-native invasive mosquitoes have been detected in the UK so far, a better system to monitor imported used tyres, in which disease-carrying mosquitoes lay their eggs, needs planning."
Despite the risk from malaria being considered low, some climate models predicted the possibility of localised infections in the UK as early as 2030.
Ticks, small blood-sucking relatives of spiders, were another danger and in the UK one species that carries Lyme disease, Ixodes ricinus, was already found in Scotland.
Another exotic tick species, Hyalomma marginatum, that transmits Crimean-Congo haemorrhagic fever virus in eastern Europe, was imported into the UK each year on migratory birds.
Currently the British climate is too cold for its survival. But as temperatures warmed the rick was likely to become established in the UK, said the researchers.