It doesn't look that bad to me. So, finish the timber first, then use coloured wax to fill the gap. You can then blend different waxes to get a colour match. Firms like Liberon, Mylands, Hafele, Konig, etc sell soft wax repair sticks to deal with stuff like this - avoid hard wax sticks as they require a ;specialist heated iron to use.
Soft wax sticks are sold for floor repairs on hardwood flooring. You need a
very flexible stopping knife to apply the wax. If new the shape corners of the stopping knife will need to be rounded off on a sharpening stone or a bit of alox paper wrapped round a piece of flat timber - takes all of 2 minutes. I'd suggest you need something like a medium oak and a pine at the very least to get a near match for an oiled or waxoyl finished oak, for example. Adding mahogany coloured wax makes your wax warmer (redder), adding antique pine makes it yellower, adding walnut makes it colder (darker/bluer)
The technique is straightforward: scrape some shavings off the sticks into your palm (in winter it's best to dump the wax into a beaker of hot water for 5 minutes first) then roll the wax into a ball, maybe the size of a small garden pea. Check the colour against the wood and add more of one or other colours to get a closer match (BTW slightly darker is always better, as is slightly yellower - your oak will darken and yellow over time). Whilst the ball is still soft and warm, smear it into the gap using the stopping knife to push it well in. Don't rub it everywhere! Keep the wax as close as possible to the gap. Leave a minute or two for it to cool then scrape the excess off the surface with the stopping knife and roll into a ball for the next section. This is why you need to blunt the tips on the stopping knife. When you are finished, take a coarse non-abrasive cloth, ideally such white Webrax - but a well washed, clean and dry floor cloth will also do - and use the dry cloth to remove any surface contamination
1in stopping knife from someone like Dulux deco centre will be under a tenner (and is great for all sorts of filling jobs), you may prefer to use a flexible plastic stopping knife (under a fiver), you can get a set of Liberon soft wax repair sticks for £7 to £10 and hopefully you'll have a coarse floor cloth somewhere. So not an expensive fix