Not hard to do.
The hardest part is communication - give a look at the ESP8266 (development boards include NodeMCU and WeMOS). More than enough pins for your needs, and built-in WiFi.
Time: fetch current time through NTP.
Room temperature & humidity: use a DHT22 or BME280.
Outside temperature: separate device; connects to the outdoor probe; acts as server where the other devices can request this data from.
LCD: a 1602 maybe?
That's all you need, except for a nice case and appropriate power supply - a 5V USB charger will do great.