How are Bath and Bristol badly connected?
In regards to Lincoln my suggestion would be to merge Lincoln-Sheffield and Sheffield-Manchester giving direct services from Lincoln to Mnachester. Also have more regular London services either through the franchised operator or open access and extend Cardiff-Nottingham to Newark and Lincoln, giving direct links to Birmingham, Derby and beyond.
I disagree with the idea to connect the Lincoln-Sheffield and the Sheffield-Manchester but I agree to have more Lincoln-London trains. They could call at stations along different lines.
1. Lincoln, Sleaford, Peterborough and London Kings Cross.
2. Lincoln, Newark North Gate, Grantham, Peterborough, Stevenage and London Kings Cross.
3. Lincoln, Nottingham, East Midlands Parkway (only some services), Leicester and London St Pancras.
Another train they could have could be a Lincoln-Plymouth calling at Lincoln, Nottingham, Leicester, Nuneaton, Birmingham New Street, Cheltenham Spa, Bristol Parkway, Bristol Temple Meeds, Taunton, Exeter St Davids, Newton Abbot and Plymouth.