The format "2020-05-25T20:11:38Z" is a common standard ISO 8601 format, it is also produced by the default Date.prototype.toString method, however it's only with a UTC (+0) offset.

The above ISO 8601 format is reliably parsed by reasonably current built–in parsers (some very old implementations won't parse it correctly), so to get a Date object:

let date = new Date('2020-05-25T20:11:38Z');

Formatting it for a fixed +1 offset can done by adjusting the Date for the offset then formatting it as required by leveraging the default toISOString method, e.g.

// Initial timestamp
let s = '2020-05-25T20:11:38Z'

// Convert s to a Date
let d = new Date(s);
// Show that it's the same date
console.log(`Initial value: ${s}\n` +
            `Parsed value : ${d.toISOString()}`);

// Create a new date with 1 hour added as 3,600,000 milliseconds
let e = new Date(d.getTime() + 3.6e6);

// Format and manually modify the offset part
let timestamp = e.toISOString().replace('Z','+01:00');
console.log(`Adjusted timestamp: ${timestamp}`);

// Parse back to date
console.log(`Parsed to a Date  : ${new Date(timestamp).toISOString()}`);

The resulting timestamp can be parsed back to a Date that represents the same instant in time as the original string (last line).

Note that the adjusted Date is only created for the sake of formatting the timestamp, it shouldn't be used for anything else.

If, on the other hand, you want a general function to format dates as ISO 8601 with the local offset, there is likely an answer at Javascript date format like ISO but local that suits. If so, then this is a duplicate, e.g. this answer or this one.

Also, there are a number of libraries that will allow specifying the formatting and timezone as separate parameters, so consider using one if you're going to do a lot of date formatting or manipulation.