Nice colors. What you're looking for is a new spindle, not axle (difference being that an axle doesn't spin, everything around it does, and on a spindle, the spindle spins). Beware of swapping out the bottom bracket; there are some compatibility issues with older Raleighs. It can be done, but it may be harder than you think. I second the idea of finding a period-correct cotterless crank. Try ebay with "vintage Raleigh cranks," or something like that.

Chrome polish (e.g. simichrome, blue magic, etc.) will work a miracle on that headbadge.

Good luck with the project.