Using prename (aliased to rename on many systems) it is possible to do this without multiple runs of mv or cp:
cd /home/user
find . -type f -name 'Loads.bdf' -exec \
prename -n 's:^./::; s:/:_:g; s:^:destination/:' {} +
Remove the -n when you are sure this is moving the files you want.
Update
To copy the files instead of moving them, the fastest way is to copy first with cpio or pax, then to rename as before. Eg (with GNU cpio):
cd /home/user
find . -type f -name 'Loads.bdf | cpio -p --make-directories destination
cd destination
find . -type f -name 'Loads.bdf' -exec prename -n 's:^./::; s:/:_:g' {} +
Note that while pax is both POSIX and LSB and cpio is no longer POSIX, cpio is still the program which is commonly found many systems.
To handle filenames with newlines with GNU find/cpio it is possible to add -print0 to find and -0 to cpio.
Update 2
The pax solution is a one-liner (not including the cd):
cd /home/user
find . -name 'Loads.bdf' | cut -d/ -f 2- | pax -rws ':/:_:g' destination
Or for any filename (and GNU find):
find /home/user -name 'Loads.bdf' -printf '%P\0' | pax -0rws ':/:_:g' destination