What to do with - suspected - self-seeded apple trees in my lawn?

[Reynard]

The new growth is generally within the boundaries of the canopies of the trees (so maybe within a 2-3m radius of the trunk); however some is distinctly removed and the garden has a distinct gradient, which made me think of self-seeding if the fruit had hit the ground and rolled.

In that case, I think your assessment of self-seeded may well be correct.

If a rootstock suckers, then the new shoots tend to be fairly close to the trunk - although my experiences of this are mainly with various Prunus species, which will sucker like it's going out of fashion LOL! (One of the reasons sloe aka blackthorn makes such a damn good layered hedge.)

 

[PaulSB]

If you have seedlings keep in mind it can take 7 - 10 years to begin fruiting. The seedlings will not, or at least it's extremely unlikely, be the same variety as the parent plant. Most apple trees require compatible crossing to produce fruit. In your situation this is a matter of pure chance. You're looking at several years to find out if what you have is any good.