Not having seem any BT fibre installs yet - our exchange is "sometime in the future"
I'm assuming this is an early install of "FTTP on demand" where AIUI they did what's in the photos.
The mass rollout currently under way uses pre-terminated cables to avoid the need for on-site fibre termination or splicing. There's a connection box fitted at the top of the pole with a passive splitter/multiplexor in it, and drop cables to each premises are plugged in at each end, with a bobbin arrangement to wind up the excess length.
What we have in the first picture the OP posted is two fibres coming in - one from the street, and the other a pre-terminated pigtail that goes into the house. Spare length on the fibres is coiled up, and the splices are in the straight tube on the left hand side. The tuft of yellow(ish) fibres sticking out of the right hand cable are kevlar to take tension applied to the cable. I would "strongly suggest" not fiddling with the fibres in there as once the sheath is stripped back, they are quite fragile.
IF there is spare length, and the hole is big enough, it would be possible to pull the pigtail back out, move the splice box, and put the pigtail in through a new hole. The hole may well not be big enough for that as it may have been a plain fibre cable without connector that was pushed through from the inside before splicing.
If you do disconnect the optical connector, do not touch the end, and
DO NOT LOOK INTO THE END as there may well be dangerous level of laser light coming out (it's not a lot of power, but it's in a very narrow beam). As a minimum, use a clean plastic bag to wrap the connector in to keep it clean until you reconnect it. Also, do not look into the optical connector on the terminating unit for the same reason.
Like Gigatator, I used to do data installs as part of my day job - but any fibre we contracted out to a specialist. Mostly because the gear needed to work with fibre can quickly add up - even if you don't do fusion splicing, but only fit connectors, you still need test gear to test and inspect your work. Also, there's a bit more to fitting and polishing a connector than there is just punching down a few bits of copper.