Make your cuts parallel to the ends of the boards and for the full width of the boards. Cut away the tongues. When replacing boards "boiler plate" the boards. It is better to take out complete section of boards to a joint rather than narrow strips. Starting with a boarded floor
Assuming that you need to remove the sections in yellow
Take the screws out, cut along the joints, remove then short bits, then make a straight cut across between the two removed sections. A circular set to about 19 or 20mm and a straight edge (piece of 2 x 1 softwood lath) temporarily screwed to the floor to guide the saw will assist, This can also be done with a multitool (and the lath is still useful for that), but will take longer. Note that the cuts are made in the middle of the joists
You will probably need a thin, curved prybar to get the floor sections up, but if the floor has been installed correctly it should have been glued down and you may well find that it won't come up without destroying it. If that is the case just replace the lifted section with new chipboard in the largest pieces you can get in there. The tops of the joists in that case will need to be thoroughly cleaned up and all old chipboard removed - a job for an old chisel and/or a floor scraper. The new repair pieces should have the tongues/grooved removed before installation (although it might be easier to get the timber yard to cut down an 8 x 4ft sheet of chipboard for you to the required sizes). Glueing the edges together is optional but I doubt that your joints will really be that good where the old timber has been cut away. Gaps can be filled with silicon sealant to give draught proofing
If it works out that you can re-use some of the boards, then you'd be best off boiler plating the joints as below:
The green piece is a piece of 18mm chipboard 150 to 200mm wide cut to fit within two joists and screwed through one already installed board (and ideally glued as well) to form a boiler plate. Alternatively the joint could be supported on a 3 x 2in softwood noggin screwed into the joists at either end as well. The next piece can be installed in the same way:
The idea is always to support the joints in shorter pieces to prevemt them from moving and cracking.
BTW, if the draught is coming up around the edges of the floor another approach is to use an expanding closed cell foam rubber strip in the joints such as Compraband in gaps between trhe floor and the wall (will require the skirting board to be removed, however)
