Are you sure that the transaction went through? any reputable exchange would clearly check for the address before sending anything.
reputable or not if you use a bitcoin P2PKH (starting with 1) instead of litecoin's (starting with L) it will be rejected by the network.
and if you use a bitcoin P2SH (starting with 3) instead of litecoin's (starting with 3) there is nothing to check and/or reject. both addresses are similar base58endocing with the same starting character which is 3 and i think the new ones are capital M!
to recover the funds, i have never done it but i assume using the seed and private keys that were used to create the bitcoin wallet should work.