If you've found an engineer who can come out immediately, why not communicate this to your landlord and ask them to authorise this engineer to carry out the works rather than the one who has limited availability?
Have you sought legal advice on your claim? This is really the best place to start, check your contents insurance policy or car insurance policy, they sometimes include legal cover. If not, Citizens Advice Bureau can help.
You won't necessarily be entitled to compensation for all the things you want to claim for. The landlord has "a reasonable period of time" to take action on a repair, which they have done if somebody came out on Wednesday, even if they were unable to fix it. A 5-day delay because boiler engineers are unable to come out immediately in a busy January will fall into "reasonable period of time." The landlord is not ignoring or refusing to carry out repairs, and 5 days does not qualify as disrepair.
You can ask for a percentage of your rent refunded for each day you've been without hot water (15% - 20% is usually considered reasonable) but no court is going to decide you are entitled to deduct your whole rent, so you need to be a bit more realistic about what you've actually suffered. Likewise time spent on the phone or what you call "private nuisance", you really just won't be compensated for this over short-term loss of hot water. You may be able to claim loss of earnings for the Friday when nobody showed up, but not the Wednesday when they did.
Is specialsubject correct that this is one of a long line of issues with your landlord? If so, as they advise, Environmental Health are your guys.