Dissolve all lines, then split them at the points. You can do it in 3 simple steps:

1. Dissolve the layer

2. At each point, create a small line, perpendicular to the initial line. Use Geometry by expression on the point layer with this expression:

    extend(
        make_line (
            $geometry,
            project (
                $geometry, 
                1,  -- use a small value
                radians(
                    90+line_interpolate_angle( 
                        overlay_nearest ('Dissolved',$geometry)[0],
                        line_locate_point (overlay_nearest ('Dissolved',$geometry)[0], $geometry)
                    )))),1,0) -- use a small value
   

3. Use Split with lines: use the dissolved layer for Input layer and the small lines from step 2 as Split layer. Now create a new field with the Field calculator expression $id.

Small, auxiliary black lines at each point, used to split the dissolved line:

Dissolve all lines, then split them at the points. You can do it in 3 simple steps:

1. Dissolve the layer

2. At each point, create a small auxiliary line, perpendicular to the initial line. Use Geometry by expression on the point layer with this expression:

    extend(
        make_line (
            $geometry,
            project (
                $geometry, 
                1,  -- use a small value
                radians(
                    90+line_interpolate_angle( 
                        overlay_nearest ('Dissolved',$geometry)[0],
                        line_locate_point (overlay_nearest ('Dissolved',$geometry)[0], $geometry)
                    )))),1,0) -- use a small value
   

3. Use Split with lines: use the dissolved layer for Input layer and the auxiliary lines from step 2 as Split layer. Now create a new field with the Field calculator expression $id.

Small, auxiliary black lines at each point, used to split the dissolved line:

The idea: dissolve all lines, then split them at the points. You can do it in 3 simple step:

1. Dissolve the layer

2. At each point, create a small line, perpendicular to the initial line. Use Geometry by expression on the point layer with this expression:

    extend(
        make_line (
            $geometry,
            project (
                $geometry, 
                1,
                radians(
                    90+line_interpolate_angle( 
                        overlay_nearest ('Dissolved',$geometry)[0],
                        line_locate_point (overlay_nearest ('Dissolved',$geometry)[0], $geometry)
                    )))),1,0)
   

3. Use Split with lines: use the dissolved layer for Input layer and the small lines from step 2 as Split layer. Now create a new field with the Field calculator expression $id.

Small, auxiliary black lines at each point, used to split the dissolved line:

Dissolve all lines, then split them at the points. You can do it in 3 simple steps:

1. Dissolve the layer

2. At each point, create a small line, perpendicular to the initial line. Use Geometry by expression on the point layer with this expression:

    extend(
        make_line (
            $geometry,
            project (
                $geometry, 
                1,  -- use a small value
                radians(
                    90+line_interpolate_angle( 
                        overlay_nearest ('Dissolved',$geometry)[0],
                        line_locate_point (overlay_nearest ('Dissolved',$geometry)[0], $geometry)
                    )))),1,0) -- use a small value
   

3. Use Split with lines: use the dissolved layer for Input layer and the small lines from step 2 as Split layer. Now create a new field with the Field calculator expression $id.

Small, auxiliary black lines at each point, used to split the dissolved line: