While I agree that you'd need modelling to get a definitive answer, I'm sure you could get a pretty good idea by just watching the congested main roads in London and seeing what most of the traffic is. As someone who goes into central London frequently, I'd say that Oxford Street is the only street that is congested and where buses make up a very high proportion of the traffic (and I believe TfL are taking action anyway to reduce the numbers of buses there). On just about every other main route I've seen in central London, I'd say that during congested periods, vehicles that appear to be either private cars or private hire vehicles vastly outnumber buses. I would therefore find it very implausible that getting a lot of the private-hire traffic onto buses could increase congestion along those routes, even allowing for some concentration of buses.
You should also consider that, in central London, even if a PHV is using back roads that are away from bus routes, you can't get very far along back streets without reaching a junction with a main road. And of course junctions are where most congestion tends to be caused. I would expect there would be very few taxi journeys that don't involve - if not travelling along a bus route - at least crossing one a couple of times and therefore contributing to junction-congestion. That's simply because almost any taxi journey that didn't cross a bus route would have to be no more than about 10 minutes walking distance - and how many people will go to the trouble of hiring a taxi for such a short journey!
One thing which hasn't come up in this debate, which is worth considering, is that of pedestrian and cycle travel.
Whilst the interaction between cyclists and buses in many cyclists eyes are to be avoided (due to the size of a bus), they have little chance to avoid cars/taxis.
If however there were measures put in place to reduce all car/taxi usage then those very same road which the taxis are driving along to stay out of the way of the busy bus routes are the very streets which cyclists would use.
Without taxis being there so much the thought of cycling becomes much more attractive to many and so now people do it. This then in turn reduces the need for so many taxis.
It was quite telling that on the Jeremy Vine show yesterday when they were talking about e-scooters that those who were the most vocal against them were saying things like "car drivers won't be used to them so they'll be at risk" and "they'll be a problem for pedestrians" and "they should be insured and taxed". As if car drivers are above reproach, pedestrians suffer no ill effects from cars and something which can be carried is going to harm the structure of the road or is going to cause damage to people or property in the same way as a 2 tonne car.
Personally as someone who cycles I'll be happy to pay a cycle tax on one condition, that cycle facilities are considerably better than they are today. That's the problem, no government will bring in such a tax as they know that people will want something in return for it and currently it would cost significantly more (over years if not decades) to provide that than it would generate.