The 12.10 slot changed to 15.30 in September 1993, with two 10 minute shows instead of one long one. Rainbow and Rosie and Jim were definitely 12.10 shows. Generally it was Rainbow Monday and Friday, with Rod Jane and Freddy on Tuesday, Allsorts on Wednesday and Puddle Lane on Thursday. At some point Playbox took over Mondays.
Home and Away followed at 12.30, but the Gulf War pushed that back to 13.20. IIRC the 12.10 slot was also taken up by news at that point - I remember mum taping Rod Jane and Freddy which we LOVED as kids, only for ITN to be on.
Rainbow was swiftly axed after Thames lost the franchise, new episodes were shown up to the end of 1991 I think, but through 1992 repeats were run before it came off entirely.
Yes, as mentioned the 12.10 slot lasted all the way up until 1993. When it began in 1972 the slot was 12.05-12.40, but in the mid-70s ran 12-12.30, with one ten minute show and one twenty minute show. That's how it stayed until 1987, but when schools programmes moved off ITV in September 1987 and they launched their first ever proper daytime schedule, the ten minute show at noon was dropped (but they were also showing them at 4pm and that continued) and the 12.10 show moved to 11.10am. This was pretty controversial (as much as these things can ever be) and loads of parents complained it was inconvenient (and meant kids coming home from school at lunch couldn't see it) so when This Morning began in 1988 they moved it back to 12.10, where it stayed until 1993*. As mentioned, Tots TV and Wizadora started in that slot in January 1993.
For the first few years, Rainbow would actually run five days a week, a couple of weeks at a time, with other shows in between, but by the end of the seventies, it had changed so the shows would run all year round on specific days. These days would be split up between the Big Four, with Thames shows on two days, and Granada, Yorkshire and Central one each. For a while Rainbow would be on both Thames' days, Tuesdays and Fridays, but in the mid-eighties was just on Friday while the Tuesday slot was home to Rod, Jane and Freddy and later the short-lived Treasure Box.
The shows you mentioned are almost correct but Mondays were Central's day, and they contributed Let's Pretend, Rosie and Jim and Playbox. As you say, Granada contributed Allsorts on Wednesdays, and Thursdays were Yorkshire with Get Up And Go, Puddle Lane and The Riddlers.
* Actually a bit later it became 12.05-12.25, because I remember the first slot for the lunchtime Home and Away on Granada was 12.25 (allowing for the regional news before the national news) because people in school got into the habit of going home to watch it and making it back in time for one o'clock.