Telephone calls on the night of the crime
For me, these calls are the clincher. Unlike the moderator, the evidence is not easy to dismiss as being tainted by corruption or bungling.
A key part of his idea was the call from Nevill to himself. In plan A, this call was to have been made by a hit man who was to be instructed which of the preset buttons to use on the cordless phone, which would retain a memory of the call, before letting himself out by the secret window which could be shut from the outside so as to appear locked. Source: Julie Mugford (creative minds who think the police planted her story are free to try and explain why they inserted this bit).
Plan A was modified with the hit man being replaced by Jeremy himself. It's not difficult to think if a couple of reasons for that but the result was a serious timing issue. He seems to have thought the time of the call from WHF to his place would be accessible in the records (it doesn't appear he was right about that, Heaven knows why not) so that time would be fixed and he would have to work around it to explain why his call to the police was not immediate. This was how he did it:
(i) he would try to call back but would find it impossible because the phone was off the hook - that could be said to consume a few minutes
(ii) then he would call Julie, feigning concern about the peculiar message received from Dad and she would advise him to call the police. Another few minutes.
(iii) then he would call the local nick (possibly wasting even more time by calling Witham police station, getting no answer there, and then calling Chelmsford - a point proved by Ann Eaton but negligently not put to Bamber by the useless cops - see page 26 of her 8th Sep statement, linked below) after looking the number up in the telephone directory. In interview, he estimated this had taken '10 minutes at the outside' to look up the number and another 5 waiting for the cops to answer the phone. I find this alone utterly incredible. A 999 call would have taken seconds. He expected to be believed in claiming that he spent 15 minutes gaffing about calling the locals when he had a vastly more efficient alternative.
Did he call the police or Julie first?
In
this statement given on 7th August (the day the bodies were found) he says he immediately called the police - see page 6.
In his
statement of 8th September, however (given under caution, after Julie had shopped him) he reversed the order. See page 6. He said he was on the phone to her for about
'two minutes not very long as I had to call the police'. On the same page he records how long it took looking everything up and hanging on etc.
In a further statement given on about 10th September, the cops pinned him down on this. See pages 15
et seq (it goes on for a few pages). They drew his attention to what he had first said and what he later said.
The problem is there is no satisfactory answer to the question. Whether he called her first or second the call looks suspicious and, being a clever person, he realised his difficulty but, not being quite clever enough, he had no way out of it and had to fall back on being unable to remember the sequence.
The next statement to consider was given by Ann Eaton on 8th September 1985. This key document contains a wealth of detail but most importantly, records her recollections of the morning of the 7th August spent at Goldhanger (Bamber's place). You can find it
here (sorry I can't cut and paste from these but they are all jpegs or something). It gets interesting around page 25.
One data point to note is Bamber telling the police that Nevill called at 3.00 a.m. He said he checked the time with his watch (and the suspicious Ann made a mental note that he must have been wearing his watch in bed

).
She observes that he did not mention the call to Julie at all (a fact I do not find at all suspicious in itself). On p.27 is the thing I mentioned before about him not speeding to the farm because he feared a trick - so that's where that comes from.
What time did Bamber call Julie?
The evidence about this is summarised by the court in his 2002 appeal, which is
here.
At about 9.50 p.m. on Tuesday, 6 August the appellant telephoned Miss Mugford. During their conversation that evening he said he was "pissed off" and had been thinking about the crime all day and that it was going to be "tonight or never". The following morning she was awoken by a telephone call from the appellant to her lodgings in London. The appellant said to her, "Everything is going well. Something is wrong at the farm. I haven't had any sleep all night … bye honey and I love you lots". Miss Mugford did not take him seriously and went back to sleep. As to the timing of this call, Miss Mugford said in evidence said that it was between 3.00 and 3.30 a.m.
A number of Miss Mugford's housemates were disturbed by the telephone call and provided additional evidence as to timing. One, Helen Eaton, had been consulted by Julie Mugford, when the latter was first making a statement to the police about it. She put the time at 3.00 a.m. in evidence but agreed in cross-examination that it might have been as late as 3.30 a.m.
Another flat mate, Sue Battersby, said that she was positive that when she was disturbed, she had looked at her clock and the time shown was 3.12 a.m. However, she pointed out that she was in the habit of keeping her clock about 10 minutes early and police checks made on the clock confirmed this to be the case. If her evidence was right and if the clock was, as the evidence suggested, ten minutes fast, the time was probably no more than a minute or two after 3 a.m.
Joanna Woad gave evidence that when she heard the telephone, she looked at her digital clock and all that she noted was that the time was 2 something. This meant that according to her clock the time was between 2.00 and 2.59 a.m. If it was at the end of that bracket, it differed very little from the time suggested by Susan Battersby's evidence.
The balance of that evidence is to put the call closer to 3.00 p.m. than 3.30 (or actually 3.36 since this is the time Bonnet noted as the time he took PC West's call and Bamber was on the other end of West's line at the time). That suggests he called Julie first and, to me, that he probably intended to use that call to help fill in the time he needed to fill as explained above.
There is a pro-innocence argument here, namely that if the call from WHF was really made at 3.00 and Bamber had called Julie at 3.02 or only a few minutes after then he could not be guilty because he couldn't be in two places at the same time. However, in that scenario, it is simply impossible to fill the time between c. 3.02 and 3.26. Bamber certainly does not offer either a credible or even a complete explanation for the delay.
Therefore, taken together with other things, he is most probably guilty.