How do they work in terms of connecting to others?
Do I just get mine done and link it and then hope others match up in time etc?
Approaching prior I don’t know but suspects are part of my lineage and asking them “are btw, would you mind getting a DNA test with me please?” seems intrusive and weird.
Hi again, sorry for delay in getting back,
I'll try and set down how I think DNA could help with this even without tracking down strangers and asking them to test (at least to begin with!

). I'm very much still in the beginners club with DNA but I have had some success with a couple of puzzles.
The most useful person to test for this would be your Dad if he is willing. If funds allow getting one for yourself too would be useful. The test kits arrive by post and after registering with Ancestry you send off the saliva sample. About six weeks later the results are available to view.
The most useful to you will be the match list. This shows everyone you share DNA with from closest to furthest. The higher the cM number the closer the relationship. The fun part is working out how you connect.
Say for example you are working with your Dad's match list. Ancestry will have assigned each match as 'parent 1' or 'parent 2' (there will also be some that are 'unassigned' - worry about those later!). This has split the matches into maternal and paternal sides but at first you won't know which is which.
Imagine then that the closest match to your dad is "Ethel" and she is assigned to 'parent 1'. Luck is on your side and you know that "Ethel" is the grandaughter of Nora Reddington's sister. Now 'parent 1' can be marked as maternal and Ancestry's system will automatically mark 'parent 2' as paternal. Anyone in this paternal group will be from Gordon's family.
Imagine now that the closest match in the paternal group is "John". You can then look to see who are the closest shared matches with him and they are "Brenda", "Tom" and "Mike". I'd start to look at any trees that these four may have - what I'm looking to work out is how they relate to each other and then work back to a common ancestor that your Dad (and thus Gordon) share with them. It will likely be necessary to build your own version of the tree that connects them.
Along the way any Fred, Rosemary, platelayers for the LNER, horse trainers/breeders, anyone born in Newark or near Arsenal football club would be getting a very hard look

. It will be quite a hard slog and will need some luck too but the circle should tighten and tighten until the origins of Gordon are revealed.
I keep going through your thread and what keeps coming up is that nothing can be found about Gordon before he goes to lodge with the Oakleys in about 1934. No birth record, no census entries, nothing. So unless he stepped off an alien spaceship it seems that he probably started out with a different name. Yours is only half the puzzle - somewhere out there is someone with a Gordon shaped hole in their family tree, and if they've already DNA tested.......
I hope that sort of makes sense. Others here on Rootschat have lots more knowledge than me and will help as you go along.
Best wishes, Tony.
Added: I don't know how I've ended up with the big space at the end!

Added: I should mention that to get full access to your match list and make best use of shared matches you will need a paid Ancestry membership and the add on "pro tools" is useful.