Nostalrius Begins - Quality wow vanilla realm (1.12)

I have tried 2 pets now and neither of them have moved past loyalty level 1 despite being well fed and each levelling once, the first pet actually started with negative training points. I don't remember it taking this long to gain loyalty.

I did receive 2 of the Hunter 'find pet' level 10 quests, could this have something to do with it?

It takes a VERY long time to level up.
It's based on how much active time they have with you - online and leveling.

There are two quests at each hunter trainer, you only need to complete one. It's just a slow process to get it to level two but after that it's easy.

stock up on meat

This might shed some light on the matter:
http://wowwiki.wikia.com/wiki/Loyalty

In my experience I feel the amount of time required is not right (is actually higher), but you a general idea.
From Loyalty 1 to 2 just requires experience, So grab a number of stacks of meat/food, tame your new pet, train him in growl, find a quiet place thick with mobs and start grinding!

For a level 1 pet you will need to feed him every 2-3 pulls as he will become unhappy very quickly.
Keep him "happy" at all times.
Once your pet dings loyalty level 2 most of the hard work is done.
Level 2 and higher requires FAR less food than 1.

loyalty 1 to 2 does not require experience. it requires constant feeding especially if the pet is high level.