Don't send that.
Firstly because it sounds like bullshit. Washing machine delivery and installation people couldn't get the machine out but a heavily pregnant woman could? Not believable and why lie anyway? Have you actually bought a new machine or is that another lie? It doesn't matter either way, the LL doesn't need to know any of this.
Don't rush out to get a new machine. Maybe your machine just needs a new door seal or a new hose. Maybe it's legs need adjusting so it sits more evenly on the floor and doesn't rattle around so much that water seeps out of the door or filter. Maybe your filter is blocked and that's why you've had a leak, have you checked it? (Water will come out when you do so have a few towels handy). Is it actually your washing machine anyway or did it come with the property (in which case it's the LL washing machine and their responsibility to fix/replace)?
Don't offer to pay. You rent through the council. You may have to tell the LL about repairs because he may be someone who prefers to do them himself or has a team of his own workmen he uses if he owns multiple properties. It's still the council paying for repairs because you rent from them. So first you need to find out if LL intends to repair the property at this point or will leave it until after you leave (or will just leave it indefinitely if it's not a safety issue, LL doesn't necessarily care too much about the state of properties) and whether you have to pay. Don't go offering it, you may not have to pay. Shit happens, you're not getting evicted for this.
What does your tenancy state about having to/not having to have tenants insurance? If it doesn't mention it then you don't have to have it. Tenants insurance isn't like homeowners insurance. Homeowners insurance is either for contents eg theft, or for buildings. Tenants aren't responsible for buildings eg roof leaks or faulty electrics causing fire. So tenants insurance is for their contents and accidental damage which the LL deems them responsible for, eg football through the window, lost keys necessitating locksmith, and possibly would cover washing machine leaks. If you're this much of a stresshead and can afford it, I suggest you get tenants insurance for the peace of mind, if you haven't already.
You don't need vinyl flooring. If it's spongy you need to replace whatever is underneath as well. It could be a really expensive job. Don't offer to pay. You probably don't have to pay, but if you do you can organize a payment plan with the council. Stop trying to fix something that isn't yet a problem. Wait and see. Your LL is the council, your rental contract is with the council.
This man owns the property, he's not your LL he's the council's LL. He's rented it to the council probably for 5yrs at a time and it's upto the council who they put in there. They'll have an agreement with him to return the property in the same condition he rented it to them, at the end of the 5yrs. My guess is he sounds like he has some get-out clause put in so he can have the property back at any time and the council have made you aware of that in your tenancy agreement. That will be if he needs to sell up or discovers you're running a cannabis farm there or something, not because he's pissed off about a washing machine leaks. He is possibly a builder or trained in building works and likes to do his own repairs. This is normal for professional LL. They often like to keep an eye on their properties and not have any Tom/Dick/Harry working on it.
Yes temporary accommodation is different. That's normal. Not come across your particular situation before. Sounds almost like a lodger agreement, but that's not right unless the LL also lives in the home with you, which doesn't sound like the case. The temporary housing I've come across was assured shorthold tenancy, same as any other regular private rental. What is your tenancy called? There are different types of temporary accommodation, and I'm not God who knows everything. Temporary gives less protection than a council tenancy of a permanent place, which is usually an assured tenancy ie basically ongoing for life, with no end date. But listen to me REGARDLESS OF THE TYPE OF TENANCY, YOURE NOT GETTING EVICTED FOR NEEDING A REPAIR so stop stressing about it.
You rent from the council, your contract is with them, you're being housed under homeless rules. Even if this LL needs the property back to carry out a repair (not going to happen in this circumstance, council tenants with house fires are sent back into the blackened mess to live there and clean up and redecorate themselves, to give you some perspective) the council still have a duty to house you so you'd be put somewhere else, not evicted and left to fend for yourself. Needing a repair doesn't make you a bad tenant. Have you any idea of the undesirables that go through the homeless system? And the way properties can get treated? The damage that can be caused? You're basically a model tenant, so stop fretting.
Just send a text saying you think the leak came from either the washing machine or sink but you're not 100% certain (because that's the truth, yes?) and you can see some damage to the floor which appears to be spongy. That's it. No need for convoluted lies, bowing down before his feet and kissing his boots in apology or offering to pay for it. You need to relax, seriously. And don't be shifting washing machines around when you're pregnant! Take care of yourself and your baby.