How long does it take to get through Security exiting Edinburgh Airport

There is no security when you land. Security is before you board an aircraft, at Malaga.
What you will have at Edinburgh is immigration (Passport control). The UK is not part of the Schengen Area, there will be immigration after you disembark at Edinburgh.
Edinburgh is a (relatively) small airport, allow 30-45 minutes for immigration.
A bigger risk is your incoming flight being late. 1 hour late is not unusual, especially for cut-price airlines who have less slack in their times.
Also, trains for the Christmas period are always very busy. Cheap tickets for Monday 23rd December have probably gone by now. Buy your tickets ASAP.

Small airport. It took about 20 minutes to go through the Border Force or passport control. Depends on how many flight arrive at the same time as yours. I think you could easily make a 1 pm train

How far are you trying to go by train?

We're traveling to Leuchars, about an hour from the train station. I would think that it would be extremely busy around the Christmas season, so we may go for the 2pm train from the Waverley Train station

That's only an hour on the train, there's one every half hour.

The Advance ticket is £8.20 but the Anytime ticket is only £7 more at £15.20.

If I were in your shoes my time (and stress) would be worth more than the difference.

I'd go for the Anytime ticket and grab whatever is going when I arrive at the station.

Thanks so much everyone for the wonderful information , it was very much appreciated

Absolutely not.
It's a very disorganised airport.
Last time I came in there, it took two and a half hours to get out.
It cost the friend who was picking me up 22 pounds for parking.
I would book my train for 2pm at the earliest.
If you're just going to Leuchars, just buy a ticket when you get to the train station.
You might not even need to book in advance, unless you want to save money by advance booking a ticket.
When you get off the tram from the airport, you still have to walk over and down into the station through crowded streets, and find your platform.
The tram does not go into the station itself, or even stop close by.
The nearest stops are either two blocks before, or one block after the station area.

I would agree that EDI seems disorganized (and crowded to the point of being nearly overwhelmed, at least the last couple times I went through there, admittedly during the busy summer season).

On arrival, I wouldn't expect long delays. Basic arrival formalities.

But upon departure...look out. Both times I departed from EDI tis past summer, check-in queues were long and slow, but the worst was going through security. It was the most chaotic, messy, slow and inefficient that I have ever seen at an airport in a developed country. Give yourself plenty of time to get through security (upon departing), and bring your patience.

I love Scotland and Edinburgh, but this is an airport that is in need of an expansion in the terminals just to catch up with the current demand. If they don't improve things, with the continuing boom in air travel, in the years ahead it's going to be a real mess.

as someone who is in and out of Edinburgh airport many times a year all I can say that in can vary from less than 5 minutes to around 30 minutes. on Saturday evening returning from Tenerife ,there were no queues at passport control and I was through in moments, September returning from Prague queues were long and it took around 30 minutes to get through.
regular buses (Ailink100) from the airport directly to the train station take 30 minutes and I have never had to wait more than a couple of minutes for one to leave.
I think what you are planning is perfectly doable as long as your incoming flight is not delayed