Slawson, who is overseeing FERA's £95,000 Stop the Spread Chelsea show garden, said: "Ash is one threat but it's important we're aware of the general risk."
He added that he does not know how much the disease will spread this year but "raising awareness" at Chelsea will help to control it.
The chestnut disease came in during 2011 and has been found at two sites, while emerald ash borer is causing damage in North America and there have been findings in Moscow but is "not in the EU yet - it's on the radar".
He said oak processionary moth, for which the Forestry Commission aerial sprayed an area of woodland near Pangbourne in West Berkshire this month, is another big issue, as are Asian longhorn beetle and citrus longhorn beetle.
When asked about legislating against importing diseased plants, Slawson said the EU plant-health regime is being reviewed and Defra has recommendations. He added that more homegrown markets could be created after recent disease scares.
"The cost of the garden is a lot of money but the cost of pests and diseases coming into the country is £1.7bn a year"said Slawson, adding that wood-boring insects cost the US £2.2bn a year.