Tusheti has a typical mountain climate with short summers (July-August) and long and freezing winters. In summer, the weather is highly variable with the sunshine quickly changing into storms or rainfall. Despite the summer months being rather richer in rainfall considering all year precipitations, there may not actually be many rainy days (commonly 4-5 in a month). Heavier rains lately come in the second half of August and are often accompanied with landslides. July, on the contrary, is rather dry and stable, while we may expect fruitful rains in May and partly in June. In any case, you should have appropriate gear when visiting the mountains in summer (waterproof jacket or raincoat, quality boots and tents). The winter tends to be dry but you will have to count with a thick snow cover.
For weather forecast you may use the Norwegian meteorological services serving Omalo, Diklo, Girevi, Dochu, or Verkhovani. Nevertheless, take into account that neither these services are 100% reliable.
Source: Accuweather.com, Yr.no