I would never recommend the Piccadilly Line to someone travelling into central from Heathrow unless they had a very heavy suitcase, weren't in a hurry and travelling to a destination with a stop on the Piccadilly Line.
The Elizabeth Line runs very frequently from Heathrow during weekdays - I've never had to wait 20 minutes for a train apart from when there were issues on the line or arriving into Heathrow very late. I really don't agree that the Piccadilly Line is the best bet for someone who has bladder issues - it's cramped, hot, slow, prone to delays en route and there's no guarantee of a seat. The Elizabeth Line is a much more pleasant experience.
The Elizabeth Line also doesn't cost any more money than the Piccadilly Line. It's only the Heathrow Express that costs more.
For someone new to London and needing to get to Russell Square, with the very real issue that getting to a toilet might be a problem, I would still recommend the Elizabeth Line.
You can go just before you get on the train - you can see in the arrivals/departures hall before you go down to the Elizabeth Line what time the next train is on the departures board, so you can wait until the last minute upstairs and go to the loo up there before you get on board. It's then about 20 - 30 mins on the tube (it always feels like 20 mins to me but I've always got my nose in a book so I may be inaccurate there!) before you're in Tottenham Court Road station, with a toilet to hand, and an easy walk to Russell Square.
However, if 30 mins is too long to be without a toilet - you might be better off with the Heathrow Express to Paddington, which is expensive, but it's very fast and there are toilets, as others have said. You would then get off at Paddington, and go out of the exit clearly marked 'Elizabeth Line' - bear in mind that at Paddington there are essentially four separate parts to the station - there's two separate sections of the underground for different tube lines that come into Paddington, there's the above ground mainline train station, and then there's a new bit outside with the Elizabeth Line - so don't follow the signs to the underground, you need to specifically follow the signs for the Elizabeth Line. You can then jump on that to Tottenham Court Road - and that only takes 10 minutes from Paddington. You can then go to the toilet again if needed there before heading on to Russell Square by foot (10 mins walk).
Good luck with it - please know as well that it's a total myth that Londoners are unfriendly - if you get lost or stuck at any point, just ask someone and they will help you!