The Russians will be unable to advance, whether with vehicles or infantry, and their logistics will be limited to good roads.
The Russians don't have to advance in Bakhmut. They are already there. Russian supplies will come in via Russian controlled territory. Ukranian resupply will also have to come in via Russian controlled territory. Roads controlled byu the enemy are never good roads. Bakhmut will just sit there and be pounded by artillery. The mud may slow advances elsewhere, but I fail to see how it does much for Bakhmut. It is a stationary target that is not going anywhere.
It does affect the ability of the Russians to fortify their lines and to resupply their forward troops, if they intend to fire on Ukrainian supply vehicles. They would not be able to bring ammo forward.
And then there will be the effect on Russian artillery. They are dispersed at the moment, but cannot stay that way. They have to displace regularly to avoid counterbattery. There are limited places they can displace to, and limited places where they can be resupplied.