More than half of the TLM’s (in full communion with Rome) are offered in mixed environment. The priest serving the TLM community serves the new rite masses too. Some laymen, including me, attend the rites alternatively (I estimate 25% the alternative users)
The possible effects of the TLM to the new rite is tabernacle facing celebration of the new rite, and usage of altar rail for communion; the wide, almost exclusive use of the the Confiteor as penitential rite and the Roman Canon; and interestingly the Leo XIII prayer after the weekday Mass . With this from the view from the pew the difference between the two rites seems minuscule.
The new rite interaction to the TLM is (in some churches) is the application of the 1964-1967 changes: no psalm 42, lesson and Gospel is read only in vernacular, either from the altar turning toward the people or from the seat and the ambo respectively; communion and conclusion similar to the new rite, no separate rite for the communion of the faithful, first benediction then Ite missa est and Placeat tibi Santa Trinitas w/o Last Gospel. This is how the Clear Creek monastery offers the TLM, this is from the 1962 Missal and by approved rubrics, so legal according to the Summorum Pontificium.
Longer term interaction will be the common calendar and common lectionary