Okay, easier to answer now with a bit more information 
Under the 2001 Code of Practice, (When there were Statements and IEPs), all the 'paperwork' had been prescribed, so there was School Action / School Action Plus / then an assessment for a Statement.
Under the 2014 Code of Practice, it now goes 'Increased Differentiation' / SEN Support / assessment for an EHCP (Education Health Care Plan). The paperwork is no longer prescribed and every school can have it's own individualised system of paperwork - some LAs decided this was completely bonkers and devised their own for the authority / others just kept using an IEP (Individual Education Plan), but many have a whole raft of 'personal plans' which are different in every school.
However every school (or Nursery, or College - that act covers children and young people from 0 - 25 yrs old) has to keep a record of what differentiation they are doing.
The IEP (or whatever the school / nursery / college calls the equivalent document) runs - as it always did under the old CoP - both before and after the EHCP (formerly Statement) has been issued. It should be on a 'Plan / Do / Review' cycle about every 3 months. So, when a difference or need is first identified, you should have Plan one. Then, 3 months later, you should review that and see if the additional support or focus has made any difference.
Lots and lots of children will have IEPs (or equivalent) / be on SEN Support (formerly SA / SA+) without ever having an EHCP (formerly Statement. I don't know current figures, but it used to be said around 20% of the school population had SEN/D, but only around 2% had needs significant to need a Statement / EHCP.
However, to answer your question, the 'IEP's start long before the EHCP. They will form part of the evidence that is considered during the assessment that the Local Authority do to see if the child needs an EHCP. The then continue afterwards, to set shorter term, SMART targets for the child to work towards.
Now - all that is to do with them accessing education.
Separate from that, is a medical diagnosis, if a child has a syndrome or condition that is medically diagnosable. that can't be done by an EP, it has to be done by a medical doctor.
You don't need a diagnosis to get an EHCP
You don't necessarily need an EHCP when you have a diagnosis, but there is clearly a strong crossover.
What would be the order of things usually, in your experience?
Do parents ask the school Senco to do initial observations/assessments first and they get an individual learning plan off the back of that? Then school and/or parents push for an EHCP? Or are they the same thing?
Normally, the school will notice that the child isn't coping. There will be meetings with parents, then will write a plan, put interventions in, then review, write plan 2, etc..
Yes, you've 'got it' in paragraph 32 of your 3rd post 
Am I to assume that the LEA will only refer for an EHCP and that the NHS will only fund an assessment if the school supports both referrals and that there needs to be several months of observation/experience first?
Or can a local GP and/or parent set the ball rolling?
It is the LA that does the assessment for an EHCP, (though a panel that should include representatives from Health, and Care, as well). Overwhelmingly the school refers to the EP, who assesses (and sort of moderates across the authority) and supports the school to apply for the assessment. Yes, there has to be evidence to back up the assessment request.
Parents can put in a Parental Request too - but they need to provide enough evidence to support that request.
Someone has to refer the child to the Paediatrician (or, I understand Clinical Psychologist in some authorities) - in my authority that can be the school, or the GP (or HV for U5s), or a SaLT or physio etc. Obviously the parent has to give permission, or take their child to the GP in the first place with their concerns. The school doesn't need to be involved if the parent takes them to the GP, but if the school and parent think the same, then a referral from the school is often a bit more powerful, as they know the child better than the GP does.
CAMHS don't get involved in an Autism assessment in our Authority. CAMHS are massively understaffed and it is very, very, very unusual here for them to be involved with a child at 6 - but I know that varies in different parts of the country.