Is mathematics becoming more applied, while engineering is becoming more math-y?
I agree with all the comments, that engineering has become more mathematical because of the rise in computers and similarly applied mathematics is growing because of its well poised position to aid in many developing engineering problems.

While I could not find any studies to back this up, I made some graphs of my own from the Web of Science.  The first two compare Applied Math
 and Pure Math.  The volatility of the Pure Math graph leads to a difficult comparison.  Similarly, I took screenshots of Engineering with Math and All Engineering. Which I believe more clearly shows a growth in engineering-math publication in the past five years or so, as compared to all engineering whose rise began earlier.

Similarly I looked into Arxiv, as it supplies a nice single source of data.  These results are for number of uploads/publications.

------------------ 2000-2004 ---- 2014-2017 ---- Growth  
App. Math --------- 577 ------------- 5456 ------- 9.45  
Pure Math --------- 253 ------------- 1399 ------- 5.52  
Eng. Math --------- 14 --------------- 476 ----------34  
All Math ---------- 14,408 --------- 102,382 ------ 7.1

While these figures are based on publications, and therefore could be caused by changes in publishing practice I think they do give decent, data-driven evidence for the claim that math is becoming more applied and engineering is becoming more mathematical.