What relation am I to my cousins?

[Loureds]

My Mum and Dad had me (obviously).

My Mum's first cousin and my Dad's first cousin, married and had children:

How are we related?

We share two sets of great-grandparents.

Thanks 😊

[John Cuthber]

I think you are "second cousins".
https://en.wikipedia.org/wiki/Cousin
 

Strictly speaking, your question is ambiguous.

You don't say who the "we" are in 
 

1 hour ago, Loureds said:

How are we related?

Do you mean you and your mum, you and your dad or you and the children of the... whatever

On a related note...

 

[Loureds]

My cousin and I share two sets of the same great-grandparents. So we both share both of our mother's grandparents and both of our father's grandparents. 

If that makes sense...

Our mother's are first cousins.

Our father's are first cousins.

[John Cuthber]

I think that makes you double second cousins.

https://en.wikipedia.org/wiki/Cousin#Multiplicities

[To_Mars_and_Beyond]

Supposing you are not trolling:

There's a measure called the kinship coefficient. It is defined as the probability that a gene sampled from person A is of the same origin as the gene sampled from the same location of person B's genome. A few examples:

• You have kinship 50% with yourself, assuming that your parents are completely unrelated. (You either pick the same gene copy twice or two different ones, so it's 50/50.)
• You have kinship 25% with your mum. To get the gene of same origin, you have to pick you maternal gene copy (50% chance) and you have to pick the copy your mum has given you from her genome (also a 50% chance).
• You have 25% kinship with your bro. Say, you pick your paternal gene copy from your genome. It's a 50% chance that you pick the paternal copy also from your bro, and it's a 50% chance that your dad has given the same copy to the both of you.

So, you see that it turns out a very interesting probability exercise. I'm sure you can work it out for the inbred family you mention. (Inbreeding is way more common than people believe.) Drawing the pedigree will help! 

[nuDAN]

I don't think there is any inbreeding here. Different grandparents, different offspring who each had a first cousin. Happens all the time. The two first cousins who married are from two completely different family lines. This person's grandparents are not related (assumption) so each of this person's parents and their respective first cousins are also not related. At least that's how I read the OP.

[Prof Reza Sanaye]

On 12/3/2020 at 11:03 AM, Loureds said:

My Mum and Dad had me (obviously).

My Mum's first cousin and my Dad's first cousin, married and had children:

How are we related?

We share two sets of great-grandparents.

Thanks 😊

In many areas of Southern Europe , we call this "cousins in distance" . .. .  ..  May be turned into English as Distant Cousins

[John Cuthber]

3 hours ago, Prof Reza Sanaye said:

May be turned into English as Distant Cousins

Not really. Second cousins, especially double ones, are quite close.
So, not only is there a correct English term for this, your suggestion is factually wrong.

[nuDAN]

John Cuthber called it. The offspring from each parents' first cousins would be YOUR Second Cousins (NOT removed).

Edited March 13, 2021 by nuDAN