You could insure your payments, but there were strict conditions about proving you were looking for work (more like the benefit ones are now, as opposed to how they were then) and they were limited. I think it was on the lines of they paid out after 12 weeks and lasted for 6 months or something. If you were made voluntarily redundant they didn't pay - if there'd been any sort of volunteering involved they didn't pay. Also, they were expensive, and when mortgage rates were high, they were unaffordable for many - maybe more so if you were likely to become unemployed.
When it comes to benefit payments, I think (but could be wrong) that if you had any savings (including redundancy payments) mortgage payments weren't covered. I think you got job-seekers' allowance for so long, then moved onto whatever it was that preceded UC. At that point they would pay towards housing costs, but again, only if you had no money. It would have been easier to get a council house, as fewer of them had been sold off, but that would still depend on points, so people with no children, or divorced dads would struggle. Homelessness was much less of a thing then though, as at least council houses still existed, although they started to get rarer as the 80s progressed. For people like the Brookside residents, many of whom had escaped 'the estate' to live on the Close, the fear of moving back after being seen to have gone up in the world would be very real. Also, the received wisdom was that anyone being made redundant over the age of 40 was on the scrap heap, and unlikely to get another job, or at least one at the same rate of pay. I suppose Paul Collins showed that side of things.
Yes, councils offered mortgages. My parents' friends had one, and it turned out to be a really good deal as it was fixed at 6%, which was very much cheaper than building society mortgages. We got our first mortgage in the 80s, and we had to be interviewed by the BS to get it. It was a full-on 'wear a suit (him) and dress (me)' job. You had to take bank statements and pay slips going back months, and they went through them all and asked about anything they wanted to. I think they even contacted your employers to make sur you weren't likely to be paid off any time soon, but I can't be sure, as it was so long ago.
We got a mortgage, and the rates had doubled before we'd completed on the house. It was scary, but we were young, and we got through somehow. House prices were rising so fast that there was no question of pulling out and waiting for rates to come down, as the mortgage multipliers were strict. Twice the male salary and once the woman's was usual, although there was some flexibility to comply with sex discrimination laws. I think it had changed to twice the higher wage (99% of the time that was the man's anyway) and once the woman's. You couldn't get any more though, so if the house you wanted had increased in price you just lost it.