Sheffield:
Every route will have changed since the city’s road layouts have changed (most significantly to accommodate the forthcoming trams in the nineties but also schemes like pedestrianisation of Leavygrave Lane meaning university routes can’t make the old left turn off the ring road, more of the city centre has been pedestrianised, the A61 road layout changed a lot around the station), frequencies of pretty much every route have been trimmed back at least a bit over the years, but the closest to recognisable would be…
51 - Lodge Moor to Gleadless looks to be unchanged outside of the city centre sections (the route used to be via Castle Square in both directions and ran via Broad Lane rather than West Street). However at Gleadless the route used to take it north west to terminate at Herdings Park, but when the trams were introduced in the mid nineties it was diverted south east to Charnock to replace the 64 instead
52/52a - slight asterisk here because the traditional 52 route at Woodhouse is that now taken by the 52a (back in 1986 there was no 52a, but that route via Retford Road gained the full route number when First/Stagecoach made peace and the “Old” Route via the top end of Beaver Hill Road is now the 52a). However, The westbound service is unchanged as far as Heavygate Road (but now extends down Walkley Road/ Walkley Lane to Hillsborough rather than turning back up Heavygate Avenue). Eastbound there’s a diversion in the city centre to accommodate High Street being made uphill only in the nineties, later diverted further when the council used the Pandemic to “socially distance” the city centre by removing buses from Leopold/ Pinstone. Overall it’s a pretty recognisable route, but not perfectly within the definition of “unchanged”
53 - Sheffield - Mansfield - halved to bihourly with a minor diversion to accommodate the road changes needed to change motorist access to the train station and the removal of the roundabout by the Showroom Cinema - inbound buses now run via Leadmill Street - but I think a time traveler would recognise it okay
272 - Sheffield - Castleton - I guess this route survives completely intact because there’s not really any alternative roads in the Hope Valley - the bulk of daytime services became the 271 for a brief period (to maintain the link from the Hallamshire Hospital to Ecclesall that the 50/30 had once done) but AFAICS the 272 number has kept running exactly the same route since deregulation beyond the road layout around the Showroom Cinema (and the dedicated turning facility at Bamford Station… Was that a change or did it always exist?)
Honourable mentions to…
…the 25 which has been the same cross city route since the late 1980s but was introduced by the independent Yorkshire Terrier a couple of years after deregulation…
…the 76 which has always run from Lowedges Road to Firth Park but the wide variety of north termini (Lane Top, High Green, Meadowhall, currently Shiregreen) are too significant to ignore - however the 76 also loses another point because, whilst the southern route/ physical terminus are unchanged, the description of that terminus has changed from Low Edges to Lowedges (I’d mention the “parallel” 75 which has had similar tinkering north of Firth Park but the change from Bradway to Batemoor gives it a weaker case)
The Sheffield - Rotherham - Conisbrough - Doncaster service has run the same route since deregulation other than those necessary for road closures etc in central Sheffield/ Rotherham/ Doncaster (and the one way at Saville/ Corby Street?) and the minor-est of diversion to serve Meadowhall, but the number was changed from 278 to X78 so it won’t qualify (there was also a 277 direct along the A630 at Conisbrough but that was scrapped)
I’m not great on cross boundary routes, I feel that there’s always been a 50 (Chesterfield - Eckington - Sheffield) and 65 (Buxton - Tideswell - Sheffield) but I can’t say with authority that they are exactly the same services and that they’ve always run without ever being renumbered or diverted (I think that the 50 was extended to Penistone for a while - or was that the 55? Certainly there was a Chesterfield - Eckington - Sheffield - Penistone route). The 65’s extension to Meadowhall probably disqualifies it but that’s beyond the main route rather than replacing any of it
A number of old service numbers still broadly relate to the route to take to certain suburbs (but the actual routes have changed a fair bit):
A “thirty” has always been a number used for the link from Waterthorpe to the City Centre via Woodhouse (Crystal Peaks itself only opened a couple of years after deregulation), but the service was the X30 for periods (now the 30 but with a couple of peak X30s and some evening/ Sunday 30a… I think for a while in the 1980s the route was an X25/X30 circular but I don’t know the route) and changed the route into the city centre between Attercliffe and the Parkway - it was extended to Dore for a few years - it has generally been serving the Hallamshire Hospital but sometimes truncated in the city centre - it’s evolved a lot whilst somehow always being some kind of “thirty”
The 57 has always linked Stocksbridge to the city centre but with a big variety of routes in Stocksbridge/ Deepcar, plus the diversion via Worrall to replace the old 65/68 and the current extension to to Moor plus a short lived Sunday extension to Holmfirth (the frequency has dropped from every twenty minutes to half hourly, back to twenty minutes, down to half an hour, hourly, currently every sixty two minutes (but there are/have been other service numbers on parallel services)
The 61 still links Hillsborough to Low Bradfield but now its a circular (with the 62) that includes Stannington and Dungworth plus High Bradfield and doesn’t run into the city centre so whilst the number has survived it’s changed a huge amount
The 82 still links Ecclesall with the city centre (The “parallel” 81 and the through service to Hillsborough were chopped for a few years but are now reintroduced, the Northern termini having changed from Middlewood to Hall Park Head to then be truncated in the city centre, re extended to Hall Park Head years later). The southern terminus isn’t the same either (the 81/82 used to alternate around the loop that the Stagecoach 88 does at Ecclesall but I can’t remember which went in which direction)
The 95 is the same route from Walkley to Brook Hill roundabout, (outbound its the same route from West Street, inbound it now runs via Broad Lane and Mappin Street due to the pedestrianisation mentioned above) but the eastern side has seen it both extended into Derbyshire and curtailed to the city centre, it currently runs to Meadowhall
The 97 has always linked Southey Green with the city centre but with a variety of northern termini (currently Hillsborough, previously Parson Cross or just Southey itself). The route was diverted away from Moonshine Lane/ Barnsley Road via both Longley and Scott Road to replace the old 22 etc, so it’s not been the same route from Southey to the city centre, and the southern termini changed from Nether Edge to Totley (and extended slightly further when new owners of the Cross Scythes pub decided they didn’t want a bus reversing in their car park, causing the PTE to build a new terminus beyond the built up areas), so the 97 is the Trigger’s Broom of bus services, continuing to link the roundabout at Southey with the northern bit of London Road (by the ring road) via Norwood Road and Wicker but having changed completely at the same time!
There’s been an M17 from Batemoor to Totley for many years (though withdrawn due to the post-Covid funding/drivers problems and reinstated), I can’t be sure that the Dore link has always been part of it as I think it used to run through to the City Centre via Fraser Road at one point? I wouldn’t want to say it existed quite as far back as 1986 and it’s not exactly the same given the lack of City Centre link, and it was withdrawn for a year or two, so not “unchanged” but it’s a pretty recognisable number
The X7 has similarly come back from the dead after funding cuts/ reinstatement, its Sheffield - fast via the Parkway - Maltby - and has changed from daytime to peak - not sure that the route at the Maltby end is unchanged - more importantly I don’t know that the number was used quite as early as 1986? Was there only the 287/288/289 all via Brinsworth at deregulation?