Before considering whether Reformed Baptists view baptism as a salvation issue, it might be worth clarifying that “Reformed” refers to those Baptist who adhere to John Calvin’s doctrine of salvation. Particular Baptists differ from General Baptists on this fundamental doctrine. Particular Baptists hold to the Calvinistic understanding that Jesus died only for the elect, and He died to actually secure their salvation, i.e., particular atonement. It is my understanding that Reformed Baptists flow out of this Particular Baptist stream.
Most Reformed Baptist churches subscribe to the London Baptist Confession of Faith (1689) as their doctrinal standards; the 1689 LBCF is essentially the Westminster Confession of Faith reworded as it pertains to baptism. Here is a partial quote from the 1689 London Baptist Confession of Faith:
Chapter 28: Of Baptism and the Lord’s Supper.
1. Baptism and the Lord’s Supper are ordinances of positive and sovereign institution, appointed by the Lord Jesus, the only lawgiver, to be continued in his church to the end of the world. (Matt. 28:19, 20; 1 Cor. 11:26)
Chapter 29: Of Baptism
1. Baptism is an ordinance of the New Testament, ordained by Jesus Christ, to be unto the party baptized, a sign of his fellowship with him, in his death and resurrection; of his being engrafted into him; of remission of sins; and of giving up into God, through Jesus Christ, to live and walk in newness of life. (Rom. 6:3-5; Col. 2:12; Gal. 3:27; Mark 1:4; Acts 22:16; Rom. 6:4)
2. Those who do actually profess repentance towards God, faith in, and obedience to, our Lord Jesus Christ, are the only proper subjects of this ordinance. (Mark 16:16; Acts 8:36, 37, 2:41, 8:12, 18:8)
The question, “Is the sacrament of baptism a means of grace according to Reformed Baptist theology?” is answered below with this declaration:
Therefore, baptism is a means of grace in Reformed Baptist theology. Source: http://www.eng.auburn.edu/~sjreeves/personal/baptism_faq.html#6
Another question discussed in the same document is “How can baptism be a means of grace in Baptist theology when Baptists assert that a person must already be saved to be eligible for baptism?” Source: http://www.eng.auburn.edu/~sjreeves/personal/baptism_faq.html#7
Basically, I understand the article to mean that baptism is not a salvation issue but results from grace experienced after the person comes to saving faith in Christ Jesus. Salvation is by grace through faith in Jesus Christ, not by works of any kind, including baptism (Ephesians 2:8-9).
I found the following quote by Dr. Kenneth Wuest, author of Word Studies in the Greek New Testament, with regard to 1 Peter 3:21 which seems to link baptism with salvation:
“The Old Testament Jew was saved before he brought the offering. That offering was only his outward testimony that he was placing faith in the Lamb of God of whom these sacrifices were a type. Water baptism is the outward testimony of the believer's inward faith. The person is saved the moment he places his faith in the Lord Jesus. Water baptism is the visible testimony to his faith and the salvation he was given in answer to that faith. Peter is careful to inform his readers that he is not teaching baptismal regeneration, namely, that a person who submits to baptism is thereby regenerated, for he says, 'not the putting away of the filth of the flesh.' Baptism, Peter explains, does not wash away the filth of the flesh, either in a literal sense as a bath for the body, nor in a metaphorical sense as a cleansing for the soul. No ceremonies really affect the conscience. But he defines what he means by salvation, in the words 'the answer of a good conscience toward God," and he explains how this is accomplished, namely, 'by the resurrection of Jesus Christ,' in that the believing sinner is identified with Him in that resurrection.” Source: https://www.gotquestions.org/baptism-1Peter-3-21.html
After I came to saving faith in Christ Jesus, I was baptised by full water immersion in a U.K. Baptist Church. Nobody said my salvation depended on being baptised. However, it was made clear by the ministers that baptism is in obedience to Jesus’ command:
Matthew 28:19: “Therefore go and make disciples of all nations, baptising them in the name of the Father and of the Son and of the Holy Spirit,”
Submitting to water baptism as a public declaration of my faith was the obvious thing to do after my sins were forgiven and I finally understood what God, in Jesus Christ, had done to save me.
Acts 22:16: “And now what are you waiting for? Get up, be baptised and wash your sins away, calling on his name.”
There was an elderly lady in my U.K. Baptist church who had resisted baptism for many years but eventually felt the need to submit to a form of public baptism – she was too frail to be fully immersed in the baptismal pool so the minister sprinkled her with water at a Sunday morning church service. It was obvious that she had come to saving faith in Christ Jesus a long time ago. Her reasons for not being baptised were known to the minister. Happily, our minister was very patient and never pressed the lady. He simply waited on the Spirit to do whatever was needed.
I have no idea what any Baptist minister would think about a Christian who refused to be baptised by full water immersion. Neither do I know if any Baptist church has issued any doctrinal statement with regard to an outright refusal to be baptised. All I know is that you can’t become a member of a Baptist church unless you have been baptised.
One thing is clear – baptism is not viewed as essential to salvation. Was the repentant thief on the cross next to Jesus baptised? No. Was he saved? Yes. (Luke 23:39-43) When asked, “What must we do to do the works God requires?” Jesus did not say you must be baptized and take communion. Rather, Jesus pointed to faith as the only “means of grace”: “The work of God is this: to believe in the one he has sent” (John 6:28-29). The rest follows.
Edit: Just found this article by the Baptist Union of Great Britain which explains (in some detail) what they believe about baptism. Bottom line is they would take a very dim view indeed of any professing Christian who refused to be baptised (for any reason)! I have to close down for this evening but will revisit your question tomorrow to insert a summary. The relevant section is from pages 23 to 27.
The document in the link is a reproduction of the 1996 booklet which is now out of print. Part of the section about baptism says the following:
The ordering of the Declaration places believers’ baptism as a secondary issue dependent on the primary convictions about the sole and absolute authority of Christ, the revelation of Scripture, and the liberty of each local church under the Holy Spirit... When someone responds to Christ both these things should follow - baptism and instruction.
Jesus commanded that disciples be baptised and his command does not expire until ‘the very end of the age’ so we must practice it. Jesus made ‘Repent and believe’ his kingdom clarion call (Mark 1:15), so linking inextricably together these two sides of commitment... Believers’ baptism testifies to this supreme experience of salvation.
The London Particular Baptist Confession of 1644 states that faith is ordinarily begotten by the preaching of the gospel and that baptism is “... an ordinance for persons professing faith or that are Disciples or taught.”
What is common to the development of all Christians are the three tenses of salvation: Christians have been saved, we are being saved (through continual growth in our relationship with God) and we shall be saved at the final appearing of Christ. Baptism is then regarded as a decisive moment in the process of being saved, whenever the process of salvation actually began (See 1 Peter 3:21).
Based on this document, it seems clear that the British reformed Baptists expect believers' baptism to follow conversion just as day follows night.