Looking through the "Janes" copy of WWII aircraft it says, "With the reconstitution of the company (BFW) as Messerschmitt A.G. the designation was changed to Me 109, the first production version to carry this designation being the Me 109E, with which the Luftwaffe went to war in 1939". "Janes" uses the Me in both 109 and 110 designations throughout, despite saying Me was introduced with Emil and no mention of Bf anywhere.
"Messerschmitt Bf 109: The Design and Operational History", by Jan Forsgren, says, "On 8 September 1927 an agreement was signed by Messerschmitt with BFW in which he would concentrate on design and development while BFW concentrated on aircraft manufacture. Messerschmitt was to retain patent rights to his designs, as a result they were two separate entities, for all practical purposes, effectively working as one company". The first reference to Me 109 comes in a statement made by Dipl. Ing. Franke in 1936 to BFW test pilot Hermann Wurster, who was about to demonstrate the He 112, “just remember that you cannot do the same crazy things you had shown us in your Me 109”. Messerschmitt being the chief designer it's understandable how the Me was given as the prefix, and possibly explains the later wide spread, though incorrect use of same.
Neither the 109 of 110 were ever officially dubbed with the Me prefix, all Luftwaffe aircraft left the factory with Bf on the data plate, and similarly the pilot notes, no mention of Messerschmitt anywhere.
It would be remiss not to mention Robert Lusser who joined BFW in 1933 and assisted in the design of the 108 and 109, later becoming head of Messerschmitt's design bureau and in charge of the Bf 110 heavy fighter project.
The link
Jhieminga kindly provides, argues that Bf and Me are both correct cites reports written in 1943 and 1944 when Me designated products were being built. I would suggest the report writer in this case just got it wrong for whatever reason. Bf products were 108, 109, 110, 161, 162, 163 (not to be confused with the Me 163 which highlights why correct designations are important) and 165, whereas there were 24 different aircraft types to carry the Me prefix – some only projects either not built or not flown. Not included is the Bf 109TL, a twin jet proposed as a Me 262 backup, comprising fuselage from the Bf 109H/BV 155B, wing from Me 409 and tricycle undercarriage from Me 309.
On this test report link you will see that the correct aircraft designations are used, Me 210, Bf 110, Bf 109, Me 209
http://www.wwiiaircraftperformance.o...-turn-time.jpg
In summation, the use of Me is not the correct prefix on those with Bf, and I don't know how "Janes" comes to its conclusion - fog of war? I'm assuming they assumed the designation changed when the company became Messerschmitt in mid 1938 and the Emil entered production in late 1938.