Arduino with multiple LEDs and timers
All Arduinos and compatible boards can work with multiple LEDs, and have multiple timers. 

For the programming capability of providing time, which presumably means a precision real-time clock, battery-backed RTC clock ICs and ready-to-use modules are ubiquitous and inexpensive. Connecting one to the Arduino is simple, and libraries are available for using the date/time/calendar/alarm functions.



The basic Arduino Uno for instance uses the Atmel ATmega328 microcontroller, and exposes 14 digital IO pins - thus supporting up to 14 LEDs directly off digital IO, more through various means such as adding an LED controller or multiplexer. Of the digital IO pins, 6 have native PWM support, usable for LED dimming.

The ATmega328 has two 8-bit and one 16-bit timer. Of these, Timer 0 (8-bit) is best not used for applications, as it is internally used by the Arduino base code for functions like  millis().



If a larger number of LEDs are to be directly programmed, or more timers are required, the Arduino Mega 2560, based on the ATmega2560 MCU, exposes 54 digital IO pins, 15 of them with native PWM support that can be used for LED dimming. 

The ATmega2560 has two 8-bit and four 16-bit timers, and again the Timer 0 caveat applies.



Without additional detail on the specific design requirements, a specific Arduino recommendation is not feasible.