Climate change may strongly modify the habitat conditions for many woody plant species. Some species could disappear from their natural habitats and become endangered, while others could adapt well to the changed environmental conditions and continue to survive successfully or even proliferate more easily. A similar process can occur within the artificial urban environment as the hitherto popularly planted urban trees may suffer from the extremities of the urban climate. However, among the planted taxa, there are species that spread spontaneously and appear as weeds in extensively managed gardens. In our study, we evaluated the native and non-native species involved in spontaneous spreading in the institutional garden of Buda Arboretum (Budapest) during the COVID-19 period in 2020–2021 when entry was prohibited, and maintenance went on in a restricted, minimal level. We investigated the correlation between spontaneously settling and planted individuals, and then performed multivariate analyses for native and non-native spreading plants for spatial and quantitative data. During our studies, we observed the spontaneous spreading of 114 woody species, of which 38 are native and 76 are non-native. Taking the total number of individuals into account, we found that, in addition to the 2653 woody species planted, a further 7087 spontaneously emerged weeds developed, which creates an additional task in the maintenance.