How will this help in deliverability, IP reputation when the tool we use to send cold email have different IP and this tool will send warmup emails from different IP.

How do These standalone warm up tools work?

For most cold email tools, how they work is that they connect to your email address via either SMTP/IMAP or a direct integration, and when they send the cold emails, they are in effect coming from your email address that you've connected and your IP address. They are not coming "from" your cold email software.

This is in contrast to some email marketing tools (like Mailchimp, Aweber, etc. that send marketing emails to your lists), where they send emails "on behalf of" your email address, but they are using their own IP addresses.

When we do email warmup for a specific connected email address, connecting to it via SMTP/IMAP or integration, we are warming up that specific email address's IP. The emails are not appearing to come "from" Boxward, they come from you. So, when your email is warmed up and you send emails later with another cold emailing tool, you are good to go.