The wax should hold up to more than one wash, but If you are not sure that the wax Isn't holding at the moment try something more durable. Collinite 476 Is a good all round protector for the weather this time of year.
With the jet wash It depends If they use TFR & If they do that will strip off the wax. Stick to the 2BM Its alot safer & If your finding It cold wrap up warm & buy some marigolds for your hands

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It'll soon be spring & normal salt free roads again

Using a quick detailer after a wash does bring the zing & beading back briefly, but It isn't meant to be durable protection.
If you use a polish on the wax It will take It off, so try topping up the wax with another coat or two.
The other thing Is, If your putting wax on In damp conditions, It may not have been able to bond to the surface properly, which will decrease Its durability.