You are given a list of osm ways' ID and you want to collect only them, along with corresponding nodes, to generate new osm file. Run this python. Will generate two files, final_osmfile.osm is the one you want at the end. You can generate selected_ways.txt by simply clicking the required ways in JOSM and pressing ctrl+c and ctrl+v.

select_osm_ways_given_wayId.py

from xml.dom import minidom
from osgeo import ogr
import os, re

proj_dir = os.getcwd()
rel_file_path = "initial_osmfile.osm";
abs_file_path = os.path.join(proj_dir, rel_file_path)
xmldoc = minidom.parse(abs_file_path)

node = xmldoc.getElementsByTagName("node")
way = xmldoc.getElementsByTagName("way")

rel_path_mid = "middle_file.osm"

with open(os.path.join(proj_dir, "selected_ways.txt")) as selected_ways:
    content = selected_ways.readlines();

node_str = '';
way_str = '';
with open(os.path.join(proj_dir, rel_path_mid), 'w') as f:
    f.write("<?xml version=\"1.0\" encoding=\"UTF-8\"?>")
    f.write("<osm version=\"0.6\" generator=\"Overpass API\">")
    f.write("<meta osm_base=\"2016-04-21T07:05:03Z\"/>")
    for n in node:
        node_str = node_str + (n.toprettyxml(indent = '  '))
    cont = [];
    for c in range(0,len(content)):
        wordList = re.sub("[^\w]", " ",  content[c]).split()
        cont.append(wordList[1]);
    for w in way:
        if w.getAttribute("id") in cont:
            way_str = way_str + (w.toprettyxml(indent = '  '))
    node_str = node_str.encode('ascii', 'ignore').decode('ascii')
    way_str = way_str.encode('ascii', 'ignore').decode('ascii')
    f.write(node_str + way_str)
    f.write("</osm>")


# Again .........................................................

xmldoc = minidom.parse(os.path.join(proj_dir, rel_path_mid))

node = xmldoc.getElementsByTagName("node")
way = xmldoc.getElementsByTagName("way")

rel_path_end = "final_osmfile.osm"

node_str = '';
way_str = '';
with open(os.path.join(proj_dir, rel_path_end), 'w') as f:
    f.write("<?xml version=\"1.0\" encoding=\"UTF-8\"?>")
    f.write("<osm version=\"0.6\" generator=\"Overpass API\">")
    f.write("<meta osm_base=\"2016-04-21T07:05:03Z\"/>")
    for w in way:
        way_str = way_str + (w.toprettyxml(indent = '  '))
        ''' collect all nodes here first '''
        nd = w.getElementsByTagName("nd")
        for nn in nd:
            ref = nn.getAttribute("ref")
            for n in node:
                if (ref == n.getAttribute("id")):
                    node_str = node_str + (n.toprettyxml(indent = '  '))
    node_str = node_str.encode('ascii', 'ignore').decode('ascii')
    way_str = way_str.encode('ascii', 'ignore').decode('ascii')
    f.write(node_str + way_str)
    f.write("</osm>")


# str_osm = '';
# for w in way:
#     cont = [];
#     for c in range(0,len(content)):
#         wordList = re.sub("[^\w]", " ",  content[c]).split()
#         cont.append(wordList[1]);
#     if w.getAttribute("id") not in cont:
#         # print "not";
#         parentNode = w.parentNode
#         parentNode.insertBefore(xmldoc.createComment(w.toxml()), w)
#         parentNode.removeChild(w)

# f = open(abs_file_path, "w")
# f.write(xmldoc.toxml())
# f.close()


# way_str = '';
# for c in range(0,len(content)):
#     wordList = re.sub("[^\w]", " ",  content[c]).split()
#     for w in way:
#         id1 = w.getAttribute("id")
#         if (id1 == wordList[1]):
#             # print "found";
#             way_str = way_str + (w.toprettyxml(indent = '  '))
#             # print (w.toprettyxml(indent = '  '))
#         # print id1
#         # print wordList[1]
#
# print way_str
# #
# doc = minidom.Document()
# foo = doc.createElement("foo")
# doc.appendChild(foo)
# print(foo.__class__)
# # xml.dom.minidom.Element
#
# print(foo.toprettyxml(indent = '  '))

selected_ways.txt

way 145430100
way 26426584
way 313727372
way 48733326
way 231051208
way 337012187
way 337012182
way 227612570
way 218081463
way 230931460