When travelling just keep the used needles in a zip lock bag (put both lids back on) and bring them back with you. You could go for a sandwich/food storage box, but I find ziplock bags of even just loose in the bottom of an overnight bag works fine - it also has the added bonus of deterring would-be searchers (while flying) from rummaging and throwing everything out - I do tell them there are sharps in there before they set to and they usually ask me to unpack for them!
I used to use a needle clipper back when I used syringes and when availability of sharps bins was rather hit and miss (and often the only ones that would be supplied were mini 0.5L which held very few syringes, so it was a major faff to have to have a new bin every week or so - there was some serious elastic potential energy stored in those cinbins to get the last few syringes in and get the lid sealed

)
Certainly where I live the council provide a medical waste service and will deliver and collect my sharps bins - I opt for 5L bins so they last a long time, which reduces the minor faff of filling out an online form to order a new one (and popping the old one on the doorstep for an early morning collection.)
Before I moved to Somerset, where the council provides this service, I used to live in BANES and at that time my GP would provide (large) sharps bins and I could bring my full ones in to be collected alongside their contingent. I'm guessing that in BANES there's now a medical waste service provided by the council too.
While from comments above that not all councils/areas provide such a service, I assume there must be some way which could be facilitated by either the GP or consultant to ensure that large volume sharps bins can be provided and collected? I also find it a bit strange that this isn't a service provided across the country.
Is there a name and shame list somewhere of which councils don't provide a service? Might be a useful thing.