You can give both plots the same PlotRange -> {{0, 2}, {0, 4}}:
Plot[x, {x, 0, 1}, 
 Epilog -> {Text[Style["y=" <> ToString[1], 10], {0.1, 0.1}]}, 
 PlotRange -> {{0, 2}, {0, 4}}]
Plot[2 x, {x, 0, 2}, 
 Epilog -> {Text[Style["y=" <> ToString[2], 10], {0.4, 0.8}]}, 
 PlotRange -> {{0, 2}, {0, 4}}]
Overlay[{%%, %}]
Instead of Overlay, you may want to experiment with a single Plot for both functions:
Plot[{x, 2 x}, {x, 0, 2}, 
 Epilog -> {Text[Style["y=" <> ToString[1], 10], {0.1, 0.1}], 
   Text[Style["y=" <> ToString[2], 10], {0.4, 0.8}]}]