Number of bits specified in standards implementation? Currently deployed RSA and discrete logarithm implementation uses $1024$ to $2048$ bits.

Hypothetically speaking if a crypto team produces a faster algorithm that moves current factoring and discrete logarithm capability to say $8192$ bits (four times today's recommendations) is there any provisions in the standard to increase the implementation to beyond $8192$ bits?

Can corporations and banks solve the issue in a matter of days?
I'm not sure if you talk about generic products or your own product, but generally 1024 bits is not considered enough anymore, and 2048 is considered borderline secure. It's still used a lot, but I'd say 2048 bits is now the bare minimum.

You're not naming any standard so we don't know if it can handle larger bit sizes. If you're talking about the RSA / DSA standards themselves: sure they can handle any kind of bit size. Implementations commonly only allow bit sizes that are a multiple of 8 or 32 bits, but that's an implementation detail.

A 1024 bit key only offers up to 80 bits of security (against the best attacks) and 2048 bit keys offer about 112 bits of security (source: NIST / keylength.com. For higher strength ciphers it is often better to look at Elliptic Curve cryptography which offers higher security for smaller sized keys. Unfortunately both RSA and EC are not secure against (large scale) quantum computers.

Corporations and banks cannot solve anything in days. To name just one issue: bank cards (smart cards) often can only handle up to 2048 bit RSA. But even if they support 4096 bits then it will be impossible to update or replace all the cards in days. Performing large scale redeployment of all keys would certainly also pose issues.