After all the talk a year or so ago of around ten minutes coming off joint line timings just four minutes is spoken of these days between Lincoln and Peterborough
I don't think 10 minutes was ever realistic, not given the rolling stock in use. I think 4 is probably erring on the side of caution, but I wouldn't have gone above 7 or 8.
There won't be much time gained between Lincoln and Sleaford even with the linespeed increase, simply because 153s acceleration is so poor. 75mph Sleaford to Spalding allows you to do it in 20 minutes rather than the currently scheduled 24, I know that from experience (although I'm not admitting to how or when!).
The unknown quantity is how much time could be gained between Spalding and Werrington. My
guess is about 3 minutes on that section.
There's also an ongoing call for increased station dwells as that's where a lot of the regular, small delays are occurring; most intermediate Joint line stops are 30 seconds, which are slowly being increased to a full minute, and both Sleaford and Spalding are seeing the standard 1 minute increased to 2 or even 3. That all bites into any time gained by the linespeed increase, and may be the real reason the "improvement" will only be 4 minutes overall: much of the time gained will be used to pad station dwells.
whilst there is the potential there to finally make the 10.15 Lincoln to Peterborough a connection from the 09.20 Grimsby to Newark at Lincoln
That's a badly timed run, the 10:15. Has 2.5 minutes pathing time approaching Sleaford to allow the 10:44 Skeg to leave, and is pretty loosely timed anyway, so it could leave Lincoln at 10:18 regardless of the Joint Line changes.
But then the inward from Grimsby's not a good runner, regularly not arriving Lincoln until 10.16 or 10.17, I think because it stands at Rasen over-time through weight of numbers boarding. In turn that means any connection at Lincoln, even if advertised to meet the 5 minutes required, is likely to be touch-and-go.
One issue I think all TOCs have is that necessary train-lengths are viewed purely through the prism of bums-on-seats. A larger train (even 2 cars instead of 1) spreads the boarding and alighting passengers with an overall reduction in station dwell time, which in turn reduces those station delays...