Children travel to Umrah every day of the year. The families who find it hardest are usually the ones who planned the trip they would have taken alone.
Where you stay matters most
Closer is worth paying for. A ten minute walk is a different thing at two in the afternoon with a tired five year old than it is at six in the morning on your own.
Timing
Do the walking early or late. Keep the middle of the day for the hotel, and treat the afternoon nap as part of the itinerary rather than a failure of it.
Prams and pushchairs
Allowed, and useful, though not in every part of the mosque at every hour. A carrier is easier in a crowd.
What to bring
Their own water bottle, snacks they already like, and something quiet to do. Nothing about a long queue is improved by discovering that nobody brought anything.