I would like to make Thai yellow curry or Panang curry but there is a severe coconut allergy in my family so I cannot put even a trace of coconut into it. I have the Mae Ploy curry pastes for both but they both call for coconut milk to be mixed in. Oh, and I am allergic to nuts so almonds (almond milk) and cashews (cashew milk) are out too :/ I have seen cream suggested but am concerned about depth of flavor
A reasonably light cream (about 15% fat) should be fine as a substitute.
It will lack the specific coconut flavour, obviously, but that's fine in this case. What's more important is the fat as a flavour carrier, and the creaminess in the texture.
A soy based cream, or even oat milk should work equally well here, too.
On the other hand, rice milk would probably not work so well, as it tends to separate quite quickly, and is generally more watery.
You may also want to try Thai country-style curry recipes. Country-style curry contains no coconut milk so there is no need to try to replicate a primary ingredient; the dish just has a different character.
Here are some recipe examples, though I have not made these.
I often cut my coconut milk in half (to decrease fat) with onion puree as is produced in Indian restaurants: youtube masala gravy. Just skip all the seasoning and use straight onion: lightly satueed then allowed to simmer in own juices for a good long while then pureed.
Silky and enough body -add cream of choice if you want richer sauce.
Does, however, require considerable advanced prep. Make a big batch and freeze flat in ziplocked bags for convenience.