What’s the future for CAT.M1 (LTE-M) and NB IoT in a 5G world?

CAT.M1 (a.k.a LTE-M) and NB IoT will continue into the future within the 5G specifications.
See also the link below to GSMA document addressing this:

https://www.gsma.com/iot/wp-content/uploads/2018/05/GSMA-5G-Mobile-IoT.pdf

Quote from page 7:
“3GPP has agreed that the LPWA use cases will continue to be addressed by evolving NB-IoT and LTE-M (CAT.M1) as part of the 5G specifications”

So for M2M the BG96 (NB IoT and CAT.M1 radios) will be relevant for a long time yet.  For higher data requirements (CAT1, CAT4 etc) will be served for many years by LTE and slowly phase into 5G after that.

In the natural order of progression of technology,  3GPP will evolve new standards for M2M use cases to migrate into the 5G evolution of Radio Access spectrum and core network.  All this is in the future,  at present BG96 (and later BG95) supports the M2M networks of LTE-M and NB IoT.

In other words, your customer does not need to worry that BG96 and BG95 (BG77) will be obsoleted with impending 5G network roll-out.
CAT.M1 and NB1 (NB IOT) will evolve into CAT.M2 and NB2 with more functions for power saving and faster data rate. Future 3GPP standards, release 15 and onwards will integrate this radio technology into 5G core networks with lower latency.

3rd June 2020: Further underscored by Telstra directly in this article.