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I have downloaded some images to be used in training set, Although I have not trained the model yet, I have already augmented the images in it.

Will it be fine if I cut paste some of those images into the validations set ? My concern is will the model's accuracy be affected, lets say: if I pick some augmented image to put on validation set, but some of the augmented variant of the same image or the original image itself is still in the training set. ?

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The augmentation is typically just a smaller variation of the original data / image. Therefore your concern is right, this will affect the accuracy value and mislead you showing a higher accuracy measure than if you would have truly independent validation data.

I would separate the data first into training and validation and then do the augmentation only on the training data set.

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