There has been some talk on TL of the sanctity of ayodhyA for jaina-s. While the shruti mentions kurukShetra as a central spot of the Arya-s, the jaina jambudvIpa-praj~nApti talks of ayodhyA/sAketa as the central kShetra of the Arya-s. They hold that several of their ford-makers were born in the city. But their shrines to the ford-makers are not specifically associated with a shrine of rAmachandra aikShvAkava.
As for the bauddha-s their shrine of a tree growing out of the tathAgata’s toothbrush is said to be the south of the city. So, while their shrines are present, they are not specifically associated with that of rAma.
However, even the buddha acknowledges that in his time the ikShvaku-s were still ruling at ayodhyA. As for the toothbrush tree the bauddha-s allege that the v1s tried to cut it down. But H tradition also records (that skanda-purANa manuscript was posted on TL) that it was the brush of rAma.
Now, whereas white indologists like to insinuate that it was a bauddha tree, stolen by v1s there is now evidence for a bauddha record older than the H one. So it could simply be either rival claims or equally possibly nAstika adoption of an older H tree shrine. Obviously the nAstika-s don’t bother about the rAma shrines.
While there are many different variants of the ayodhyA-mAhAtmya of the H AFAIK they agree that there was a kShetra associated with the rAma-janmasthAna. At least a subset of the versions mention shrines of other heroes/deities like multiple vAnara-s hanumat, sugrIva, nala, nIla etc, vibhIShaNa and lakShmaNa. Most of these are also gone today.
Both H and jaina texts mentions a peculiar guardian deity ayodhyA-rakShaka called mattagajendra. While today some people say he is a rAkShasa related to vibhIShaNa (may be son) there is no such connection in the Skt texts. Who was this mattagajendra? Notably he is accompanied by other deities vighneshvara and piNDAraka. These appear to be gaNa-s of kubera who were survivals from an old kaubera system of yakSha-s that is only faintly remembered by the H and jaina texts.