Margaret,
Apologies, upon rereading, my reply was too blunt.
However, from my understanding, you have a system installed that is less than 6 months old. At the time of handover to you, you believed it worked properly. Oddly, the installation engineer ( or someone after installation) told you to leave the immersion on all the time. That on reflection is the alarm bell that they could not get the hot water to function on a stand alone basis. My interpretation of that was that they had got confused with the wiring when installing Nest. That is probably the case, but I may be wrong.
The pictures of your installation show both poor workmanship and planning of pipework routes. whilst the latter is not a cardinal sin - it shows that the installer at the very minimum did not think through and plan how he /she was going to deliver a quality and visually attractive installation. There does not appear to be a return filter fitted ( which may be a requirement for the Valiant boiler warranty).
The crossed pipes are either a sign of poor installation planning or reversing flow / return post installation.
Whatever the issue, it should be corrected by the installer. I woukd strongly advise you to write to the installer, setting out all the issues and give them 30 days to correct the installation. Thereafter ( assuming you are in England /Wales consider taking the case to the small claims court - it will cost you £55, which is refundable when you get an award. If the installer does not respond to the case, judgement will (in 90% of cases) be in your favour. Enforcing a judgement is quite straightforward
Is it fixable, of course it is, the question is always how much will it cost, to identify the issues and then correct them.
Was the engineer who came this week the original installer or a “new” set of eyes?