According to the Janka wood hardness tests chart, Elm is not among the strongest woods:
https://ejmas.com/tin/2009tin/tinart_goldstein_0904.html
1 inch thickness on the thinner fork limb should be OK, but I would personally cut the fork limbs shorter (see image), as this will remove a large part of the risk factor in terms of structural strength. An inner fork width of 4 cm (1.57 inches) should be fine for ammo up to 9 mm (.38 cal.), and a reduced fork height should allow for flat bands with a suitable draw weigh for that purpose.
I would in any case lock the forks between vise clamps and push against the handle with fair force: if it handles that, you should be OK. Even if the outside of the wood looks fine, this is a good way to ascertain the state of the inner wood sections of the forks. There can be interesting surprises sometimes, and not the kind you want on a natural slingshot.
You do not want broken sections of forks flying back at you. :hmm:
I would stick to a simple wrap & tuck groove placed on the front side of natty frame facing away from the shooter to maximize fork limb strength.