No. He took out a mortgage on the main house, owned by the "Robert Lord Trust", and likely financed the purchase of the second house with it.
Nick and Kayla bought the Silent Hills house in the 2000s. The current house wasn't bought until 2018 (after the deaths of both Louis and Peaches, so the money probably came from Celeste).
A kiwi discovered the existence of the mortgage quite a while back. I can't remember why people were poking around on property records but we found out "Lady Rackets" was Kayla from the title on the original house. I don't recall a mortgage on the original house being found at the same time. It's possible that mummy cleared whatever was owing on that at the same time as the new house was purchased.
The extensions and remodelling Nick was talking about doing to the main house would have been expensive. Land is cheaper in rural areas but labour and materials aren't. A pretty average house extension costs about $150,000 here and Nick had grand plans which likely included remodelling and renovating as well. I can easily believe that the money was originally borrowed to carry out those plans (you do have tell tell the lender why you want the money) and then became just a pool of ready cash when Nick realised that he knew fuck all about building and it wasn't going to be nearly as easy as he thought. He didn't think the ride was going to end, so why not?
Nick didn't grow up surrounded by enormous wealth, but he did grow up surrounded by success and his ego would make him believe that success was guaranteed once he found his niche. I doubt it ever occurred to him that maintaining success is a different beast entirely, because he never stuck at anything (he freely admitted getting bored with things easily and moving on to something else). And, of course, since Celeste inherited from the Owens estate, Nick has always had a (massive) safety net, which he's actually acknowledged.
Even without the drugs, Nick was still on a collision course with reality. He liked the trappings of the rich and famous lifestyle far too much for someone who was neither rich nor famous. I have no doubt that Celeste paid for all the charitable event tickets and charitable donations which were made in Nick and Kayla's names, but there was plenty of evidence of other financial excess which Nick didn't have the income to sustain. Hell, only a month or two back he was still buying wagyu beef.