If you use epoxy resin, it will not be necessary to use screws. In theory, you could just use epoxy without the timber but it would be very expensive and it will slump out unless you use perspex to hold it in place as it cures.
RepairCare have some videos on their site. Be aware that they recommend using the wood "hardener" specific to the particular epoxy product (which bumps up the cost). Timbabuild only have two types of epoxy (1 hour cure and 4 hour cure), whilst they do sell an epoxy wood hardener (EWS)
http://www.chemfix.co.uk/products/t.../timbabuild-epoxy-wood-repair/timbabuild-ews/ it can be used with either of their epoxy fillers.
If you shop around you can get the (well made) Timbabuild gun for about £25. The rubbish plastic RepairCare gun is about £30 and the decent quality one is about £50. The Timbabuild fillers are three to four times cheaper than the RepairCare ones but RepairCare is stocked by more local suppliers where I live in west London (actually, I have never seen Timbabuild branded products in any shops, ever- that's the power of marketing).
RepairCare might be the better option if you wanted epoxies that cure at very low temperatures (eg winter), in most other cases I would recommend Timbabuild (so long as you are happy to wait for delivery). There are other brands available, but I have no experience of using them.
As an aside, the 1 hour Timbabuild (ERC10) is much easier to sand and more flexible than the 4 hour EHB60 but the former is limited to 10mm thickness. If splicing timber I would use the ERC10. It slumps a lot though so splicing in the timber means that you have less sanding to worry about when compared to just using filler. BTW when I say easier to sand I use the term relatively, machine sanding with 60 grit paper (and dust extraction) would be advisable.
Oh, wear gloves and mix the products thoroughly (otherwise they will not harden). I use off cuts of wood/tile/laminate/etc to mix up on, alternatively I use the plastic board that was part of the overpriced RepairCare introductory kit (£125) and then use decorators wipes to clean the board and filling knives.
Please do provide feedback once you proceed with whichever product you go for- it helps other people that search for similar solutions.
Given the cost of the products, when splicing wasn't viable I have been known to use small pebbles to bulk out the filler, thereby reducing the amount of epoxy needed.