I did a similar calculation but came back more optimistically (although still on scale of a year to two).
You only need count over 18s as very few cases in this group. It is believed by the UK experts to have a lower fatality rate of 1% and likely lower admission rate due to asymptomatic infections (estimates at 16-75%). Ventilator capacity can increase with stopping surgery and CPAP/other measures can be used in some cases. Treatments developed later should improve efficiency of treatment. Extra ventilators should be manufactured and help ease burden. I think 80% is too high, likely more 60% given an R0 of 2-3. Finally, the chronically unwel elderly are unlikely to benefit from ITU care significantly and would never qualify. With these exceptions it gets down to 1-2 years depending on assumptions.
No. People in UK: ~66,000,000
Est. proportion of people who will get sick: 80%
Est. proportion of sick people who require intensive care: 5% (in Italy it's 10%)
So that's 60,000,0000.80.05 = 2,640,000
Number of ICU beds in the UK: ~4100 (90% occupied, but let's ignore that for now)
Length of time ICU required: at least 1 week (probably longer, but let's ignore that too)
So that's 2,640,000 / 4100 = 644 weeks or 12 years that we would need to spread the cases over in order to have enough capacity.