Creating a timer in python

I'd use a timedelta object.

from datetime import datetime, timedelta

...
period = timedelta(minutes=1)
next_time = datetime.now() + period
minutes = 0
while run == 'start':
    if next_time <= datetime.now():
        minutes += 1
        next_time += period

You can really simplify this whole program by using time.sleep:

import time
run = raw_input("Start? > ")
mins = 0
# Only run if the user types in "start"
if run == "start":
    # Loop until we reach 20 minutes running
    while mins != 20:
        print(">>>>>>>>>>>>>>>>>>>>> {}".format(mins))
        # Sleep for a minute
        time.sleep(60)
        # Increment the minute total
        mins += 1
    # Bring up the dialog box here

See Timer Objects from threading.

How about

from threading import Timer

def timeout():
    print("Game over")

# duration is in seconds
t = Timer(20 * 60, timeout)
t.start()

# wait for time completion
t.join()

Should you want pass arguments to the timeout function, you can give them in the timer constructor:

def timeout(foo, bar=None):
    print('The arguments were: foo: {}, bar: {}'.format(foo, bar))

t = Timer(20 * 60, timeout, args=['something'], kwargs={'bar': 'else'})

Or you can use functools.partial to create a bound function, or you can pass in an instance-bound method.

Your code's perfect except that you must do the following replacement:

minutes += 1 #instead of mins = minutes + 1

or

minutes = minutes + 1 #instead of mins = minutes + 1

but here's another solution to this problem:

def wait(time_in_seconds):
    time.sleep(time_in_seconds) #here it would be 1200 seconds (20 mins)