Contacttower
Let me give you a decent reply then.
"Assuming one is flying around the packed airspace of Southern England "
It is busy only in some places, and mainly below 1500ft or so. In past 10 years, 11 UK mid-airs of which 10 below 1000ft and 1 reportedly at 1800ft.
Few in GA would fly at those levels if going somewhere "fast". Most serious VFR pilots fly as high as they can, clouds and airspace permitting. In the absence of weather/airspace one would fly at 4000+ft as this gives you calm air even on hot summer days. The best MPG (typical IFR tourer, non-turbo engine) is at 7000-10000ft.
"trying to get words in edge ways"
This is the way the PPL is taught (high radio workload, inside leg measurement supplied to every ATC unit within 2,000nm radius) but one doesn't actually do this when flying for real. You can go non-radio in Class G - no point in calling up any ATC enroute unless they can offer you a radar service, and if you get that, and give them your routing (A-B-C-D using navaid waypoints, not village names) they tend to leave you alone, with implicit MATZ transits etc. They are busy enough dealing with pilots who don't know where they are going or are non-transponding so nobody knows how high they are.
" while trying to stay aware of your position with nothing but DR and the map."
Why navigate WW1-style when there is no need to? Get a big moving map GPS, and your nav workload falls by at least 90%.
"At what speed does this simply become too difficult? At 100kts it's possible, but at 250kts say would it still be practical or would it just be a matter of time before you ended up busting airspace?"
I can't speak for dead reckoning (I don't do it) but the RAF seem to get about OK at higher speeds, allegedly with just maps, headings and timing. Mind you, they are the cream of the cream of the young men (those that don't make the grade used to become navigators and end up in the CAA
) and don't compare with civilian pilot recruits at all. They also have good ground backup and - due to generally poor fuel endurance - fly carefully preplanned missions. Some of them do have GPS; in fact any military plane that goes anywhere near any real action needs to have proper nav.
250kt at 3000ft would be a piece of cake, with a GPS - just follow the magenta line, manually or on autopilot. I've seen 220kt ground speed and it wasn't really noticeable; the ground below moves just a little faster than usual. I often wish I had 250kt IAS, to cover distances. Obviously one doesn't fly the circuit at 250kt.
Poor visibility is something else. You can fly legal UK-PPL VFR with 3000m vis, which is more or less total haze ahead. I don't see anybody dead reckoning in that, other than at a very low speed and low level. Very hard work.
Regarding the 250kt limit, this is 250kt IAS. With the TAS multiplier (in a 250kt IAS descent through/from FL100) and some tailwind, your GS could easily be 300-350kt and you would still be legal.