No, the A45 didn't take any of the Peterborough line alignment.
At Northampton it crossed under the A45 then ran south of the A45 alignment all the way to Wellingborough - south of Billing Aquadrome and Great Doddington.
At Wellingborough the A45 crossed the line's formation at roughly Whitworth's mill - the line was heading north into Wellingborough, the A45 is East West.
From Wellingborough the line's formation is north of the A45 running north of the lakes initially and then through the middle of what is now Stanwick Lakes Country park. It then headed north of Ringstead and into Thrapston.
The A 605 which is the Thrapston - Peterborough road did take some at Barnwell and I think around Oundle.
You can check all of this using current OS maps.
Again - a very limited amount. The only bit used was from the bridge under the M1 at what is now Junc 15A to somewhere around Blisworth so about 1 mile.
From Blisworth the SMJ ran north of the current A43 formation skirting north of Tiffield. Having passed through Towcester the railway formation crosses the A43 at right angles on the Towcester by-pass section, just after that the SMJ split with a line which headed north and on through Moreton Pinkney and Byfield to Fenny Compton. The southward route headed to Banbury joining the LNW branch from Buckingham at Cockley Brake Junction, but the SMJ route to that point ran well to the north of the path of the A43 all the way to Banbury. The railway ran north of Abthorpe and Wappenham, through Helmdon and south of Greatworth. In fact the SMJ ran north of Brackley where the current A43 heads south-west to join the M40.
And the reason why Northampton - Peterborough isn't comparable to Bedford - Cambridge, is because Bedford - Cambridge is actually about linking Oxford and Milton Keynes to Cambridge. Bedford-Cambridge is simply the missing link on a route which is otherwise complete. Northampton - Peterborough has completely gone and you're making (what I believe to be an incorrect) assumption that there is a traffic demand between Northampton and Peterborough which the rail network could tap into. The reason the A45 / A605 is busy isn't people travelling between Northampton and Peterborough - it's people connecting onto the A1 - the number of lorries on that bit of road is a giveaway to that. Peterborough relatively speaking is not a key destination for people from Northampton nor vice versa.
I stand corrected about the re-use of trackbeds (and, as an aside, the bit of the A43 that goes under the M1 is on the trackbed of the, briefly electrified, Northampton- London route via Blisworth; my mention of the SMJ was in response to an earlier post). It is the case, however, that the A45, like the railway, follows the river Nene and the road and rail are certainly close enough to achieve similar travel opportunities and I was certainly misled, near Stanwick, by what I thought was an old lineside structure.
My mention of Northampton - Peterborough related to correcting the suggestion earlier that populations along the route east of Wellingborough are insignificant. I wasn't arguing for its re-opening and I'm sorry if I gave that impression - I personally travelled on the line but I have no wish to re-create my youthful adventures on the railways - time moves on and I think the car is far more convenient. The County is just as populous and economically successful as its neighbours but most of its population don't even consider the possibility of using rail, for reasons I tried to explore.
Perhaps I was being too obscure but I was trying to develop the argument that there was as good as, or better, a case based on population and connections for re- opening this line than the others being considered and, bearing in mind in the real world it was a no-hoper (a suggestion you seem to support) where did that leave the others?
My comparison with Bedford-Cambridge was part of this logical argument. By the way, I don't see that this section is the missing link between Oxford and Cambridge - last time I looked over a bridge parapet near Verney Jct there's something around missing there, too!