1) child born in the UK does not automatically receive British citizenship by birth. For a child to be registered as a British citizen at birth, one or more parents will need to hold British Citizenship or 'settled status'

2) 

It is mandatory to apply for Indian passport of the Child alongwith registration of birth. The registration of birth must be done at the Indian Consulate within one year of Child's birth.

Parents of the child have to register birth online at indiancitizenshiponline.nic.in/  including uploading the picture. A print out of the registration done online needs to be self-attested by both parents and the same has to be attached with the miscellaneous service application seeking registration of birth at the Consulate.

3) A child born outside India on or after 03.12.2004 shall be a citizen of India by descent, if either of the parents (father or mother) is an Indian citizen at the time of birth of the child and the birth of the child has been registered with the concerned Indian Mission abroad within one year of the birth and the parents of the child have to submit a declaration that the child does not hold passport of another country. Approval of the Central Government is necessary in all such cases where the request for such registration is made after the expiry of the said period i.e. after one year.

4) A visitor visa for a maximum of five specified places may be issued for a longer period up to two years with multiple entries to –

(b) a Pakistan National married to an Indian national

5) 

person who is married to an Indian national can apply for Indian citizenship under the section 5(1)(c) of the Citizenship Act. The clause (1)(c) of section 5 which provides for ‘Citizenship by registration’, reads, “…..The central government may, on an application made in this behalf, register as citizen of India, any person not being an illegal immigrant …. who is married to a citizen of India and is ordinarily resident in India for seven years before making an application for registration”.

As per MHA rules, for such applicants, mandatory documents include a copy of a valid foreign passport, a copy of valid residential permit (RP)/Long Term Visa (LTV), evidence of his/her spouse’s Indian nationality and a copy of marriage certificate issued by the registrar.

1. The child born in UK would acquire the citizenship of UK.  The status of the child would be similar to that of the children born to the parents who are not legally married.

2. It will be the citizen of UK.

3. The child is now a UK citizen by birth.

4. The child born to Indian father and Pakistan mother  but took birth in UK would be an UK citizen, if you want to register its birth in Indian High commission in UK, you may approach the authorities concerned  and comply with the necessary formalities i this regard.

Unregistered Indian Children Born in the UK are Stateless and Entitled to British Citizenship. The High Court has ruled that a six year old Indian child who was born in the UK is stateless and therefore entitled to British citizenship.

5. You both are not married to each other hence your partner cannot be considered as your souse without marrying you. 

There is no specific visa called Spouse visa for India. For any foreigners married to Indians, you can get an Entry visa (X) through any of the FRROs in India. This visa doesn't give you any rights to be employed or do business, but you can stay for a long term in India on this visa.

8. e-Visa is granted to a foreigner whose sole objective of visiting India is recreation, sight seeing,
casual visit to meet friends or relatives, attending a short term yoga programme, medical
treatment including treatment under Indian systems of medicine and business purpose and no
other purpose/ activity. This facility shall not be available if the person or either of his / her
parents or grand parents (paternal or maternal) was born in, or was permanently resident in
Pakistan

9. Generally people do get it within six months to one year but MHA is not bound to give it within any specified time limit. There is no exception in rules for people from Pakistan.” A person who is married to an Indian national can apply for Indian citizenship under the section 5(1)(c) of the Citizenship Act.

10. All applicants of Pakistani origin who hold dual British-Pakistan nationality must apply for an Indian visa on their Pakistan passport. ... The processing time for visa applications received from persons of Pakistani origin is 7-8 weeks or more.

You can marry your partner under provisions of special marriage act in India . She can come to India under visitor visa for the marriage 

2) on basis of marriage certificate apply for visa for your wife and son 

3) apply for registration of son birth with Indian embassy in UK . Approval of central govt is necessary as birth has been registered after period of one year 

4) you can marry your partner in UK and register your marriage with Indian embassy under provisions of foreign marriage act 

5) your son can be Indian citizen as father is Indian citizen 

You need to legalise your relationship with your live in partner 

register your child birth with Indian embassy 

1. Unregistered Indian Children Born in the UK are Stateless and Entitled to British Citizenship. 

2. You have not married your partner  hence she cannot be termed as your legally valid wife, therefore she may not be entitled to visa as a spouse even if you apply for a visa on that grounds.

3.  If you are not married to her then she is no one to you.

You may say that you both have been living together for years, but that may not entitle  her any rights on you as her husband neither you can claim rights over her as your wife.

 For marriage under special marriage act you can apply for the same in the Indian embassy at UK itself by complying with the necessary formalities. 

You cannot bring her or your child without valid papers identifying them as your wife and child. 

4.  Under Schedule 2 of the British Nationality Act 1981, a person who is stateless can be registered as a British Citizen if he or she:

• was born in the UK
• is and has always been stateless
• was under the age of 22 on the date of the application
• has lived in the UK for the 5 years prior to the making of the application.

Your child may be eligible for British citizenship if the:

• Child was born in the UK.
• Child's parents are Indian Nationals.
• Child has not registered with the Indian High Commission under indian Citizenship act 1955.
• Child has never travelled abroad.
• Child should not be entitled to any other nationality.
• Child hasn't applied for permission to register as Indian National from Central Govt. India.
• Child was born on or after 3rd December 2004 in the UK.

A child born outside India on or after 03.12.2004 shall be a citizen of India by descent, if
either of the parents (father or mother) is an Indian citizen at the time of birth of the child and the birth of the child has been registered with the concerned Indian Mission abroad within one year of the birth and the parents of the child have to submit a declaration that the child does not hold passport of another country.

Approval of the Central Government is necessary in all such cases where the request for such registration is made after the expiry of the said period.

5. No comments.

You are in a quagmire or in  a hazardous situation.

Therefore you may first try to register your marriage through Indian embassy under the provisions of special marriage act and then  look for further formalities in this regard.