If a Catholic marries in a non-catholic service, or by a Justice of the Peace they may not under normal circumstances receive communion until they have gone to confession and attempted to regularize the marriage.
The exceptions to the rule are that a bishop may for serious reason allow the person to marry in a non-catholic service, usually a Catholic priest present to wittness for the Church, the conditions may include that a Catholic service be held sometimes in private. In the UK the State requires that a person who is qualified preform a civil service, and then a Religious Service may take place. Most all Anglican Clergy are certified to preform the civil service, and usually do in either the rectory or the sacristy prior to the Church wedding. Select clergy of other Churches in the UK are also qualified to do so.
To stretch the answer out even further, the Church requires her subjects to follow both civil and Church law, in order that the State also recognize the marriage, even though it usually has different standards. So if the State requires a license, or civil ceremony, the Church allows it, as long as the Church service is also preformed.
It must be remembered that the priest does not really marry the people. What he does is wittnesses the marriage, offers the blessings of the Church, and assures that the marriage is conducted in a proper manner. If no priest can be obtained in a reasonable time, a marriage may take place between two people with two Catholic wittnesses, and is blessed when a priest is availible. This was more common when the US and other countries were mission terretories, and a priest only came through once or twice a year.