My guess is that, if there is time left in the options the broker will wait, but if not, then the long put will expire worthless, and the trader will simply end up long the shares. If that's the case, then the put spread carries the same risk as a short put, is this correct?
The long option can always be exercised, it doesn't need liquidity.
But what can happen is that the underlying expires between the options, in which case the short option will be assigned, and the long option will expire worthless. There is some time after expiration, before the settlement date, to call the broker and request that the long option be exercised, but you really shouldn't let things get that far.
So, yes, the maximum risk of a credit spread is not always limited to the size of the spread.
Assuming that short put (a) and the short put of the bearish vertical spread (b) have the same strike price and expiration then the loss on each position will be the same at the long strike less the premium paid for the it. Example:
(a) Sell an Oct $50 put for $2.
(b) Sell the Oct $50/$45 put vertical for $1.25 (+ $2.00 - $0.75)
The maximum loss for both positions is $3.75 at $44.25. Above $44.25 short put (a) loses less or makes more than vertical spread (b). Below $44.25 short put (a) loses more than vertical spread (b). Being assigned early does not change these numbers.
If assigned early and you allow the long call to expire then the risk in short put (a) will always be 75 cents less than the shares bought via vertical (b) since you received a larger credit. If you do not want the directional exposure of the stock below $45, make your adjustments before expiration.