Bollocks would be a good word to start with. Complete and utter as useful adjectives.
<<I'd thought that the Nimrod was always planned to have a better radar (Searchwater) and that the ASV21 fit in the MR1 was always viewed as interim.>>
This is true but the Nimrod entered service, with trials, in 1970-71 whereas the Seachwater was about 7 years later.
<<But how fair is it to compare the rest of the kit?>>
The active tracking system was a direct bolt in from the Shackleton - Mk 1c Sonobouys, active and passive, the former with a range of 3500 yards and a slow data rate. Even when introduced it was out moded for modern, and not so modern, nuclears. The passive would work if you could induce the nuclear to cavitate. They were not daft so they would not cavitate unless you got a torpedo close enough to count.
<<Presumably the MAD was similar, or even the same,>>
The Shack did not have MAD and the Nimrod did not have Autolocus.
<<but how about the ESM?>>
The Shack had a joke of an ESM, Orange Harvest, the sparkplug on the top. It could detect and give a bearing for S or X band radar, on a good day with a following wind.
The ARAR/ARAX in the Nimrod was ex-Atlantic and a brilliant piece of kit operated by equally brilliant operators. It was enhanced by the creation of the ESM Data handbook, a database created almost single handed my MAEOp McConnel (may be Connel) at ISK against much opposition from NATO who did not want NATO radar parameters released to the Maritime force.
<<And how about sonics processing>>
The Mk 1c Sonics system and the Jezebel LOFAR systems were integrated into the central computer system to the extent that bearings could be transmitted and displayed. Later software mods allowed the Tac Nav to see which sonobuoys the Wet Team was monitoring. Once this Mod was introduced it was remarkable how often the Wet Team was not monitoring sonobuoys adjacent to the buoy in contact.
<<and the central computer?>>
This owed its lineage to the Chieftain Tank for which an 8k 16-bit computer was adequate. It used core storage - something that Babbage might have recognised - a small wooden frame with sets of parallel wires in a grid. The intersections with a small iron washer. As it was made of iron it was rusty!
<<And did the Shack MR2/3 have anything like the same nav kit?>>
The Shack had Blue Silk/GPI 4/GIVB whereas the Nimrod had Decca 67(?)/Inertial/and a modern compass system. The simple details elude me. The doppler was a bigger version of the type fitted to the Dominie. The IN was a steal from the Blue Steel. Extra panel lighting courtesy of the Lancaster George VI pillar lamps.
Both had Loran as a long range fixing aid but that in the Nimrod was as fitted to the Britannia.
The Nimrod had a Kollesman Sextant that could be used to slew the IN which was not the world's best. The Shack used the Mk 9.
Weapons wise both aircraft used the Mk 30 and Mk 44 torpedoes, carried Lindholme Gear, and 5 inch flares. The Nimrod dropped the Marker Marine through its sonobuoy launcher. Instead of smoke floats the Nimrod was fitted with a Retro Launcher that fired a French made smoke float at aircraft groundspeed. It would fall more or less where the aircraft fired it.
The Mk 30 was a passive torpedo and well obsolete even before it was fitted to the Nimrod. It had a search speed of 12 knots and an atack speed of 18 kts. Most conventionals could outrun it. It was however very quiet and could 'sanitize' an area for 20 minutes.
The Mk 44 was no racing snake either with a noisy search speed of 30 kts. It ran for 6 minutes. Even if it caught its target it is quite likely that it would not have worked as we only got certified and effective fuses shortly before it was withdrawn from service.