Alternative-switching multiple busses
The simplest would probably be to put some 10-bit 3-state line buffers after the 4017s. Something like 74ABT827 (working with 5V supply - or 74LVC827 if you're working with 3.3V). Wire the four Y1 outputs of the four 74xx827 chips together, the four Y2 outputs together, etc... up to the four Y10 outputs. You'll get the combination of digits you need on the resulting 10-signal bus.

Now, you have to ensure only one 74xx827 chip is enabled (using the OE1/OE2 input) at a given time (or there will be contention), and that each chip gets enabled for the requested time, cycling one after the other.

For this, a possibility is to use another 4017, wire its Q4 output to its RST input, and wire each of its Q0-Q3 output to each of the OE1/OE2 inputs of the 74xx827 chips through an inverter (don't forget the inverter!). Then clock this 4017 with whatever signal that has the appropriate frquency on the board (the 1Hz output from U4, maybe?). 

Note: the 4017 may not initially have its Q0 output active on power-on. So you may have to wait for a few seconds for the display to show up if the active output is somewhere from Q5 to Q9. To prevent this, wire the OR-combination of all outputs from Q4 to Q9 to its RST input, rather than just Q4.