What is the process of removing an automatic excommunication? If a baptised Catholic committed sins which would have excommunicated him from the Church, what would be the method for getting back?

Receiving communion would not be allowed without reconciliation. Would a priest forgive those sins if they were all Mortal?
There are indeed some sins which cause excommunication simply by being committed. Such excommunications are called latae sententiae excommunications (in Latin, more or less, &quot;[excommunications] of a hidden sentence&quot;). Acts which can incur such an excommunication include:

a physical assault against a bishop or the Pope;
procurement of an abortion;
attempting to celebrate the Eucharist, even though not an ordained priest.

The Catechism of the Catholic Church states:

Certain particularly grave sins incur excommunication, the most severe ecclesiastical penalty, which impedes the reception of the sacraments and the exercise of certain ecclesiastical acts, and for which absolution consequently cannot be granted, according to canon law, except by the Pope, the bishop of the place or priests authorized by them. In danger of death any priest, even if deprived of faculties for hearing confessions, can absolve from every sin and excommunication.

(paragraph 1463)
In particular, the Code of Canon Law provides:

Canon 1354 §3 If the Apostolic See has reserved the remission of a penalty to itself or to others, the reservation must be interpreted strictly.
Canon 1356 §1. The following can remit a ferendae sententiae or latae sententiae penalty established by a precept not issued by the Apostolic See:

the ordinary of the place where the offender is present;

Canon 1357 §1. ... a confessor can remit in the internal sacramental forum an undeclared latae sententiae censure of excommunication or interdict if it is burdensome for the penitent to remain in the state of grave sin during the time necessary for the competent superior to make provision.
§2. In granting the remission, the confessor is to impose on the penitent, under the penalty of reincidence, the obligation of making recourse within a month to the competent superior or to a priest endowed with the faculty and the obligation of obeying his mandates; in the meantime he is to impose a suitable penance and, insofar as it is demanded, reparation of any scandal and damage; however, recourse can also be made through the confessor, without mention of the name.

Here's what that means:

If a particular excommunication is specified as only being removable by the Pope (for example, if someone assaults the Pope), then officially the person needs to confess to the Pope. In practice, the Pope would probably depute the local bishop to hear the confession and remit the penalty.
In other cases, the local bishop himself (or a priest deputized by the bishop) would need to hear the penitent's confession.
In any case the confessor would impose an appropriate penance which would have to be observed.
After that, with sins forgiven, the person would be innocent and once more in good standing with the Church.

As far as whether a priest would absolve these mortal sins: Catholic priests are obligated to absolve from sin any penitent who seeks reconciliation and genuinely appears sincere, regardless of what their sins are:

If the confessor has no doubt about the disposition of the penitent, and the penitent seeks absolution, absolution is to be neither refused nor deferred.

(Code of Canon Law, Canon 980)