PhineasC
No, you do not have to be heading 240, this applies generally to aircraft being vectored from the north.
Generally, however, air traffic control (ATC) will give you a heading from which you will intercept the ILS centreline. This heading is normally about 30 degrees different to the final approach track, so either 240 or 300 degrees for westerly operations, but this 30 degree can vary between 20 and 40 degrees from experience, as a function of wind, traffic and radio congestion (it can sometimes be turn right heading 300 degrees and establish from the south!)
Furthermore, the ATC instructions will normally be something like "callsign .... turn right/left heading 240/300 degrees, descend 3000 feet QNH 1020, report localiser establlished". The point at which you establish on the localiser will usually get from ATC "callsign .... descend with the glideslope" followed at the same time, or shortly thereafter, by speed instructions and an instruction to contact LHR Tower.