The first Tu-160M2 aircraft (serial number 08-04, Pyotr Deynekin) was demonstrated publicly during president's visit to the Gorbunov Aviation Plant in Kazan. The aircraft reportedly conducted its maiden flight in late December 2017. Also during the visit the ministry of defense and PAO Tupolev signed a contract to build ten Tu-160M2 aircraft. The cost of the ten-year contract is said to be 160 billion rubles (about $3 billion).

It is not clear if Pyotr Deynekin is included in the initial order of ten planes. It appears to have been assembled from an unfinished plane left over from the Soviet days. However, it may be considered Tu-160M2 if it is eventually equipped with new avionics. It seems that the Tu-160M2 project is largely a plan to re-start the production of Tu-160 bombers "from scratch."

UPDATE: Things may be a bit more complicated - Pyotr Deynekin is a Tu-160M, not M2 (h/t DS). Which seems reasonable - it is very much an old Tu-160 design. It gets an "M" when it gets some new avionics. The contract signed in Kazan also appears to cover the construction of Tu-160M, but at some point after 2020 these planes will be converted to Tu-160M2 (and presumably new ones would be built as M2). Other sources say that Tu-160M is a "deep modernization" of Tu-160, which means that these are not new planes. This would mean that the contract would cover only modernization and not a new construction. In this case, production of Tu-160M2 would be covered by a different contract.

UPDATE: It appears that the new aircraft will be built as Tu-160M2 after all. At least, in tenders announced by Aviastar (here and here, h/t AS) the new plane is referred to as "izdeliye 70M2".

The other strategic-bomber project - PAK-DA - may take its first flight "in mid-2020s." No significant changes there compared to the earlier plan.