I want to send alerts via SMS.
I want to know how this can be achieved without Twilio extension.
There was a similar question before - "How can we send SMS to a user using ThingWorx?": https://community.ptc.com/t5/ThingWorx-Developers/How-can-we-send-SMS-to-a-user-using-ThingWorx/td-p...
The Article - CS345079 - Changes to ThingWorx extension support model deals with where to get Twilio, From the conversation you are not interested in using that solution.
When you have a Cell Phone number if you know the service provider you can construct an email message and direct it to the cell phone. You need to go to a website such as https://email2sms.info. This type of site provides information which identifies the gateway to send the message. With the gate way information you can construct an email address <PhoneNumber>@<SMSgateway>
I do not know what information you are referring to when you say "phone number's mail id"
You need to use a service such as "Freecarrierlookup.com" This service take phone number and performs a lookup and returns information on the carrier and SMS Gateway Address. I am sure there are other web sites and/or services which provide this information.
There is also the information provided by email2sms.info. This site provides a JSON or CSV lists of provider and there gateways. If you are collection the cell phone number from the user you can request that they provide a service provide and then use that to lookup the gateway.
To the best of my knowledge there is no universal gateway.