I think with the information provided above, we are all speculating as to why the a/c very almost crashed well off to the side of the runway at ema.
Not being qualified on the older variants of 737 I am unable to comment on what warnings or indications were available to the crew at the time as to the Localiser deviation, however here's a few thoughts of mine, and those that I have heard:
1) How about if both autopilots were not angaged for the approach? The aircraft would fly initially down the ILS as normal, the first warning sign available to the crew would be an amber single channel flag on FMA, and then also the lack of FLARE armed indication. Miss these warnings and the a/c will not complete an autoland. Try and do a go-around in this situation at minimums, press TOGA, you'd expect the A/c to pitch up and do a dual channel go-around, what happens? The autopilot disconnects, you continue to descend and eventually after a few microseconds realise it is your job to manually go-around. It is feasable for this to happen in some 73 variants due to the procedure for a go-around differing depending on if there is two autopilots engaged or not.
2) The Localiser indication would have almost certainly shown more than 1/3 dot deviation (regardless of the cause!), calling for a go-around pretty sharpish to avoid turning into a 140kt grass cutter! The decision time is minimal, not even seconds! Add a long shift, tired crew etc it's possible that it may have been missed. No commercial pilot would be able to tell you they maintain 100% alertness at all times, and have never been a little tired on the last approach of the day!
I think we're all going to learn a lot from this incident when all the facts become available, and it just reiterates the fact that you really do need to stay sharp ready to deal with the unexpected.
Well done to the crew who undoubtedly did a good job to get on the ground safely in BHX, faced with a second diversion with an aircraft that had such damage, there would have been incredible pressure on the crew that night diverting to BHX.