If a husband wants to recommit to Catholicism but his wife does not want to convert, will they be considered to be adulterers?

Conversion to Catholicism does not make one an adulterer. Other factors would come into play here. If either the person wanting to convert or the Spouse was previously married and divorced, then the Church might consider the marriage as adulterous, but the matter would need to be brought before the marriage tribunal for a decision. If neither party had been previously married, the Catholic church would not consider this an adulterous relationship.

This is a pastoral advice question, and each case is best answered by a Catholic priest who is knowledgeable in all of the particulars of the case.

"Reversion" is straightforward. Once a Catholic, always a Catholic. However, the falling-away from the Church is contrary to Canon Law:

209 §1 Christ's faithful are bound to preserve their communion with the Church at all times, even in their external actions.

This is something which may be confessed and absolved, and that — with the appropriate penance applied by the priest — is sufficient to be reconciled to the Church.

With regard to the marriage, the husband has contracted a marriage with an unbaptised person. The Catechism of the Church recognises that this presents particular difficulties:

1633 In many countries the situation of a mixed marriage (marriage between a Catholic and a baptized non-Catholic) often arises. It requires particular attention on the part of couples and their pastors. A case of marriage with disparity of cult (between a Catholic and a nonbaptized person) requires even greater circumspection.

1634 Difference of confession between the spouses does not constitute an insurmountable obstacle for marriage, when they succeed in placing in common what they have received from their respective communities, and learn from each other the way in which each lives in fidelity to Christ. But the difficulties of mixed marriages must not be underestimated. They arise from the fact that the separation of Christians has not yet been overcome. The spouses risk experiencing the tragedy of Christian disunity even in the heart of their own home. Disparity of cult can further aggravate these difficulties. Differences about faith and the very notion of marriage, but also different religious mentalities, can become sources of tension in marriage, especially as regards the education of children. The temptation to religious indifference can then arise.

1635 According to the law in force in the Latin Church, a mixed marriage needs for liceity the express permission of ecclesiastical authority.¹³⁵ In case of disparity of cult an express dispensation from this impediment is required for the validity of the marriage.¹³⁶ This permission or dispensation presupposes that both parties know and do not exclude the essential ends and properties of marriage and the obligations assumed by the Catholic party concerning the baptism and education of the children in the Catholic Church.¹³⁷

1636 Through ecumenical dialogue Christian communities in many regions have been able to put into effect a common pastoral practice for mixed marriages. Its task is to help such couples live out their particular situation in the light of faith, overcome the tensions between the couple's obligations to each other and towards their ecclesial communities, and encourage the flowering of what is common to them in faith and respect for what separates them.

1637 In marriages with disparity of cult the Catholic spouse has a particular task: "For the unbelieving husband is consecrated through his wife, and the unbelieving wife is consecrated through her husband."¹³⁸ It is a great joy for the Christian spouse and for the Church if this "consecration" should lead to the free conversion of the other spouse to the Christian faith.¹³⁹ Sincere married love, the humble and patient practice of the family virtues, and perseverance in prayer can prepare the non-believing spouse to accept the grace of conversion.

As the Catechism shows in its references, the marriage is subject to Canon Law and the Tribunal, as well the civil authorities:

1059 Even if only one party is Catholic, the marriage of Catholics is governed not only by divine law but also by canon law, without prejudice to the competence of civil authority concerning the merely civil effects of the same marriage.

A baptised person should have been confirmed before marriage (Canon 1065); but being presented with a fait accompli makes that impossible.

Canon 1086 covers the case explicitly:

1086 §1 A marriage is invalid when one of the two persons was baptised in the Catholic Church or received into it, and the other was not baptised. ^A

Canon 1125 lays down steps which are necessary for validation of a mixed marriage:

1125 The local ordinary [that is, bishop] can grant [permission] if there is a just and reasonable cause. He is not to grant it unless the following conditions have been fulfilled:

1. the Catholic party is to declare that he or she is prepared to remove dangers of defecting from the faith and is to make a sincere promise to do all in his or her power so that all offspring are baptized and brought up in the Catholic Church;

2. the other party is to be informed at an appropriate time about the promises which the Catholic party is to make, in such a way that it is certain that he or she is truly aware of the promise and obligation of the Catholic party;

3. both parties are to be instructed about the purposes and essential properties of marriage which neither of the contracting parties is to exclude.

Part of satisfying (1) may well be to receive confirmation if that hasn't already happened [Cf Canon 1065].

In order to validate a marriage, it will probably be necessary to make the vows which the local Bishops' Conference have mandated. This can normally be done in a relatively private ceremony with the minimum of fuss.

Without a valid marriage, then yes, the couple will be seen as "living in sin," and this could restrict the Catholic party's access to the sacraments. However, the sin is the conjugal act, so living in continence awaiting the validation of the marriage would be the good thing to do, rather like an engaged couple. The Catechism again:

2350 Those who are engaged to marry are called to live chastity in continence. They should see in this time of testing a discovery of mutual respect, an apprenticeship in fidelity, and the hope of receiving one another from God. They should reserve for marriage the expressions of affection that belong to married love. They will help each other grow in chastity.

The "sin" certainly isn't adultery.

2380 Adultery refers to marital infidelity. When two partners, of whom at least one is married to another party, have sexual relations - even transient ones - they commit adultery.

In this case, the two people are civilly married to each other, so it can't be adultery. I suppose that canonically it's fornication, since the marriage isn't canonically valid. However, as the couple are married to each other, even that is doubtful:

2353 Fornication is carnal union between an unmarried man and an unmarried woman.

This comes down to the adage "If you think something is a sin, it probably is." Note that I wrote will be seen as "living in sin" — that is, to live together without a valid marriage may cause scandal. This is something where local customs become important and the local clergy's advice will be particularly helpful.

¹³⁵ Canon 1124
¹³⁶ Canon 1086
¹³⁷ Canon 1125
¹³⁸ 1 Corinthians 7:14
¹³⁹ Cf 1 Cor 7:16

^A Note that the online version of Canon 1086 has not been updated to remove the phrase and has not by a formal act defected from it, which was removed by the Motu Proprio Omnium et Mentem in October 2009.