I'm afraid that I feel this is a skill that is a young man's territory. I used to be able to learn a new song in a matter of hours and be able to play it a year later with virtually no practice in between. Now, that is a distant memory. I have stopped even trying. I just use a song book and a music stand when I am performing (after all our classical cousins have always done that, haven't they? (Though, before anyone says anything, I do know, personally, a phenomenal professional pianist who keeps hours of music in his head – he’s in his 30s).
As a young man, I kept several hours of repertory in my head - my own and songs I loved, but would never have learned to play in public. Many of these songs are still available to memory (though open to the occasional stutter or lapses) - these include whole albums by Bob Dylan and Joni Mitchell that I can still sing by heart. BUT - anything recent has little chance of making it into the permanent memory banks.
From 30 to 40 I learned to Speak Spanish, Catalan, French and Italian (I use the first two every day and I become quite efficient at speaking French whenever I am in France. Having only visited Italy on holiday a few times, my Italian is nothing like as fluent, but I can still get by. Then in my forties I did a Portuguese course and spent a week in Lisbon. Although, I can still read the language, I can no longer speak it, because by then the memory cells were not retaining things as they used to.
This year, to prepare myself for a visit to Germany over the summer, I spent six months with a German course and grammar books, watched German films with German subtitles, etc. In Germany, I was pleasantly surprised that I got by (we were largely in rural areas where very few people spoke English).
However, now, four months after returning, I seem to have forgotten even the little I learned.
There is probably the contributing factor of just how much information passes across our visual cortex every day, nowadays. I read threads here, I have a Facebook account, I keep up with world events, I translate texts, prepare classes and mark student essays for a living, etc. All of this must take its toll on a 58 year old brain.
I read somewhere that in a single month Internet users are exposed to the amount of information our ancestors would have processed in a whole lifetime. Like Paul McCartney, I am certain my Memory is “Almost Full”.
So, in short, unless you cut out most other aspects of our Internet age and concentrate solely on learning songs – and your cerebral cortex is under 40 years old – I doubt there is too much to be gained by worrying about not being able to learn songs quickly by heart. Like poor eyesight and aches and pains, we must just have to get used to it and enjoy what we still can do... while we still can – as the old song goes “Don’t worry! Be happy! And play your Ukulele!” – Though part of that quote might be my memory playing tricks with me.